Lunaport: Math, Mechanics Transport

07/30/2021 ∙ by Paul C. Kainen, et al. ∙ Georgetown University 0

Issues for transport facilities on the lunar surface related to science, engineering, architecture, and human-factors are discussed. Logistic decisions made in the next decade may be crucial to financial success. In addition to outlining some of the problems and their relations with math and computation, the paper provides useful resources for decision-makers, scientists, and engineers.

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1 Location

One possible location for a major space-transport hub would be planet Earth but it seems likely that the extremely high-power propulsion systems of space-going vehicles will be too dangerous for a terrestrial site to be feasible. So we shall assume that the spaceport is actually not on Earth.

Earth orbit is the closest option, as indeed is the case with the International Space Station. But the ISS has a crew of, say, a dozen, while for a major facility, there are thousands of people involved in the various technical specialties required for operations. It does not seem likely that a null-G environment would be comfortable for non-Astronauts. Even the Astronauts don’t stay more than a year or two because of bone-loss due to weightlessness [82].

Another issue is maintenance and labor cost. Carrying out the simplest task in an orbital station requires extensive training due to free-fall conditions. A lunar environment, with 1/6-G, will also require some practice, but the huge difference is that even low gravity provides a fixed vertical orientation. In contrast, in null-G (i.e., free-fall), the axis of symmetry is lost! Not knowing up from down can be quite psychologically unsettling, even to the extent of producing vertigo and physical and mental distress in a person (space sickness). Astronauts are able to cope, but only with lengthy and very expensive training.

Being in space, there are no damping factors, so large and small-scale vibrations could echo back and forth. Transient load-differences on an orbital station would have to be carefully monitored, lest unwanted parasitic vibrations be induced which could destroy the station. Large bridges have fallen because of such structural resonances. So free-standing, undamped structures, as in orbit, could have fundamental problems with stability under heavy use.

One solution, for the comfort of travelers, crew, and support personnel, might be to have the station rotate. In that case, a separate system of small vehicles, rather than simple tubes, would be needed to carry travelers to their interplanetary ships, thereby adding an additional stage: earth to station, station to ship, then the ship takes the passenger on the long-haul voyage to another planet (or asteroid or space station). It is the station-to-ship portion of the total journey that is made more complicated if the orbital station rotates.

Also, a rotating orbital station might still have issues with dynamical stability, and it is not obvious how one might handle an emergency situation involving major structural damage. A damaged rotating structure might tend to disintegrate, flinging various compartments (possibly with intact air and filled with people) into a myriad of different directions.

Hence, we shall assume from now on that the facility is on the Moon and call it Lunaport - as it was named in the Star Trek series [122]. Arguments for such a moon base are in [135] and [126] (Artemis Base Camp, ABC).

Granting the necessity of Lunaport, where should it be located on the Moon? The first choice is whether to build on the lunar surface or below it. If there exist “lava tubes” [22], then they would offer a shielded location with no risk from radiation and meteorites and greatly reduced construction cost. However, for a spaceport, it seems necessary that the basic facilities be on the surface unless we develop technologies capable of first excavating a large region (or using an existing crater) and then covering it with a secure shield that can open doors to allow access for spacecraft.

So we shall assume a surface location. There are then various choices: polar vs. equatorial, Earth-facing side vs. hidden side, Northern vs. Southern hemispheres. For polar and equatorial locations, one considers meteoroid frequency, proximity to useful materials (e.g., water-ice in shadowed craters), local topography (e.g., sunlit peaks), and so forth. Earth-facing or not might be determined differently for factories and hotels.

Current consensus is for location at the S. Pole; see, e.g., [135, 33, 119, 125]. The Artemis Base Camp design makes sophisticated use of local terrain features near Shackleton Crater. Lunaport is what ABC may become when there are scientific and commercial missions on 5 planets and 15 moons and asteroids.

The existence of special points in orbital systems is important to power generation and communications, which will be an integral part of lunar operations. The five Lagrange points (L1, , L5) are defined for suitable pairs of orbiting bodies - such as earth and sun or moon and earth - and they allow a small object to remain in a fixed position with respect to two larger orbiting bodies; see [97]. The stable Lagrange points are those points in the orbit which are 60 ahead, L4, or 60 behind, L5. As early as 1974, the Princeton Univ. physicist, Gerald K. O’Neill proposed using the moon-earth L5 as a location for a space colony [100], see also [103] for some historical perspective. Stability means objects remain in place without correction.

More exactly, L4 and L5 correspond to maximum points in potential energy, while L1, L2, and L3 correspond to saddle points (maximum in one direction, minimum in another), so none of the five is strictly stable [97]. But as a point leaves L4 or L5, it becomes subject to the Coriolis force which causes it to stably orbit the Lagrange point as centripetal force balances gravitational.

The other three Lagrangian points require occasional correction but allow objects to maintain position with minor expenditure of energy. The L1 point for earth and sun is between the earth and sun and so would be an ideal location for a massive solar energy station. It is currently used for a solar observatory, the SOHO satellite. The L2 point is the future location of the James Webb Space Telescope. This Lagrange point keeps earth between it and the sun and so is well-suited to deep space viewing. The L3 point is where “counter-earth” would be - the point opposite the earth on the other side of the sun. It and the moon’s darkside are good locations for SETI’s radio-telescopes.

The L1 and L2 points for the moon’s orbit about earth make ideal locations for lunar telecom as the moon keeps one face toward earth and the other away, so L1 handles the earth-facing hemisphere and L2 handles the darkside.

Additional locational problems include Low Earth Orbit (LEO) (getting crowded [138, 87]), geostationary (may become so), graveyard orbits [88], satellite internet constellations [112] and finally Deep Space Early Warning satellites [139, 140]. A further development might be a sensor array able to function like the early-warning systems now in place for tsunamis but watching for large meteors, comets, and asteroids. An interesting alternative might be “cheapsats”; i.e., inexpensive , transient satellites, in orbits which decay so the sats burn up in the atmosphere (like reflecting confetti) or fall into the sun.

2 Health and safety

For any domain with no atmosphere, solar coronal discharge and cosmic rays pose variable and chronic risks, respectively. A solar storm of the Carrington-type that melted earth-based transformers in the mid-19th century would be very bad if it hit earth today [34] and might be even more lethal on the lunar surface. One reason to go to space is to better observe the sun to prepare for such events. On the moon, the risk might be somewhat mitigated by localized magnetic fields in certain areas [86]. There are more recent studies and simulations which raise new possibilities for the “strong lunar crustal anomalies” [89], [90] and hence there may be regions with inherently less radiation risk.

Satellites and instrumented platforms have ridden out such electromagnetic storms (like ships at sea) if they can shut down fast enough. Thus, research on the lunar magnetosphere as well as the stability of its lithosphere will be vital to achieve a safe environment. However, suitable shielding and “grounding” of the electronics is certainly a priority since otherwise an EMP (electromagnetic pulse) event would pose a potential vulnerability.

Unlike solar storms which are variable, cosmic ray levels seem to be quite predictable. It should, however, be noted that we have a rather short base-line regarding the stability of the cosmic ray levels. It is believed that they are fairly constant unless there is a nearby nova or supernova in the galactic neighborhood. But, for all we know, there are streams of much higher levels through which the solar system may occasionally pass.

Humans residing for an extended period on the moon will require multiple shields to be protected from dangerous radiation and meteoroids; this imposes a significant design constraint on Lunaport’s architecture (see below).

A recent article on lunar urban design [47] mentions the radiation issue and also discusses a risk particular to the moon: dust consisting of small, sharp, abrasive particles (unsmoothed by air or water friction), ionically-charged and sticking to things.

Obviously, meteoroids pose a risk for lunar structures. Before attempting construction, it is vital to know the rate at which a mass meteoroid hits a square km. One might think that meteor risks are well callibrated but this does not appear to be the case. An article [34] from 2016 claims that each day a 14 square-km patch collects about one gram of extra-lunar material. But the claimed mean value is subject to further validation and is of limited utility until the statistical variance is better specified. In fact, intensity is not constant; there are several annual meteor showers, and some of them, like the Perseids, last for a couple of days. Depending on the size-range in a given meteor stream and its width, a portion of the moon exposed to the stream might experience orders of magnitude more impacts than the average. Thus, the geometry of astronomical phenomena is important in assessing safety.

The latest (2021) data from NASA claims that speeds of 20 to 72 km/sec (45,000 to 160,000 mph) are the typical approximate range for meteoroids hitting the lunar surface [81]. A 50 kg meteoroid hit the moon at about 17 km/sec (38,000 mph) [79] and larger meteoroids on the moon appear to be more common than previously thought [13]

, further underlining the need for more careful investigation. One can estimate the frequency of larger meteor falls using seismic sensors left on the lunar surface, also optically, as such impacts produce a visible flash, though the type of material involved in the collision is related to the percentage of impact energy converted into light

[136].

In [134], two of the NASA scientists who authored [136] estimated that there are 300 to 400 hits per year on the lunar surface by meteoroids of mass at least one kg. As the total surface area of the moon is about 4/3 that of Africa, roughly 40 million square-km, one can expect that a given 100 square km area during a given year has one chance in a thousand to be hit by a meteoroid of mass at least one kilogram. Hence, in a thousand years, it has about a 63 percent chance of being hit. The trouble is that there may be many more meteoroids in the 100 to 1000 gram range. Even at the low end of mass, if the rock is traveling at 70 km/sec (a hypervelocity well beyond what can be done in the lab), it is difficult to imagine how to stop such meteoroids. (One possible solution is proposed in §6 on Architecture.)

The View from Space

In mechanical engineering, a desired structure is constructed upon a stage and analyzed there. But the “fourth wall” of engineering is the human who must use the device.

The science of designing machines to fit the comfort and operating parameters of the human body is called ergonomics, and it is currently practiced “in the breech” (that is, not at all) in motor vehicle design. Cars now have large blind-spots for the driver, and the electronic displays and unexpected human voices in these vehicles can be extremely distracting. Cruise control is equally dangerous as it temporarily turns the driver into a passenger. The physiologic nature of the human - reach and reaction time - impose critical limits.

In the body, one finds obvious bilateral symmetry combined with interesting anti-symmetries such as gait (arms vs legs) and action/perception (cortical hemisphere vs hand/eye), and these too must be reflected in design.

Machines intended for use on the factory floor are only an issue ergonomically to their operators but those used in transport of humans and other living beings must be more carefully considered - as when metal and flesh collide, things go poorly for the organic side.

Aside from service personnel and scientists, who else might want to live on the Moon? If the voyage from Earth can be smooth enough, perhaps many individuals with health and mobility issues would make the trip, happy to live with 1/6th their weight. However, such travel would likely be one-way, as the re-adaptation to full gravity for such a person might be fatal. Some people might even travel to the moon to have heart operations if low-G significantly improves survival.

3 Energy sources

I believe the sun will provide ample power, not just for Lunaport but also for the earth itself, and in fact the easy availability of solar energy will be a substantial part of the raison d’etre for space travel.

Power from the sun is currently obtained either directly, via photovoltaic devices, which convert sunlight into electricity, or indirectly by using radiant heat to boil a fluid to run a turbine and thereby a dynamo. The first is called PV and the second SP (solar power). For the second method and sometimes the first, the sun’s light and heat are concentrated using lenses or mirrors. Both technologies are undergoing rapid development.

Solar collectors could be placed at the L1 point with respect to earth’s orbit about the sun, where it is always noon. According to NASA’s diagram [97], the moon might possibly occlude the straight-line joining this L1 point to earth, but in practice, it would likely be safer to beam the energy to satellites or the lunar surface, and to never aim at the earth itself. Possibly, automated factories can be co-located with the power source, so that among other products, batteries could be filled with the sun’s energy very inexpensively.

Initially, solar energy farms on the moon may be needed, as lunar industrialization is a natural first step [100] before building large-scale space facilities such as a solar energy array at a suitable Lagrangian point in the earth’s orbit around the sun. When the orbital collectors are built, using micromechanically controlled mirror arrays, an efficient design would have the individual units convert, via PV or some other process, the visible light and ultraviolet (UV) energy directly into electricity, while the heat energy is concentrated at the focus of the faceted array to do work there.

Another source of energy on the moon could be nuclear (fission). As the ecological hazards disappear, nuclear power becomes much more feasible. But meteoroid danger suggests that, as with solar power, the power should be produced by a number of independent smaller units.

Being in space and having a huge supply of solar (or nuclear) energy might allow some seemingly far-fetched methods to generate even more energy. Successful fusion power and other, less well-known approaches could turn out to be practical, given the abundance of solar energy and vacuum.

For instance, the Casimir effect is the creation of a repulsive or attractive force between two metal plates sufficiently close to each other in a vacuum. The surprising effect has been observed in experiments and has been shown to be potentially expoitable by MEMS [16]. If exogenous power (e.g., from the sun) can move two plates sufficiently close to initiate the effect, then perhaps piezo-electric modules could use the force to produce an electric current. Such a battery might last a very long time and could be ideal for the vacuum of space. Another use for a super-abundance of solar power might be the manufacture of metallic hydrogen [14]; see the next section.

Perhaps not just the light but also the electromagnetic field of the sun can be used to generate energy in the manner of wind and water power. A conductor moving through the solar magnetic field should generate a current. The solar wind and solar flares could be harvested by a deep space, widely-distributed network of small satellites with MEMS, a “wide-net of microsats”, and such wide-nets apply to astronomy (Very Long Baseline Interferometry, VLBI), power collection/transmission, and communications, cf. [112].

4 Propulsion

How to get from Earth to Moon is a critical choice which will affect all other design issues. Currently, most space flight involves rockets with a variety of propellants. The figure of merit is specific impulse, defined as the ratio of impulsive force to mass per second of fuel expended; units are in seconds [98].

Typically, a fuel and an oxidizer (stored as liquids) are mixed together and ignited. Hydrogen/oxygen combination (LH2 with LOX) is the most powerful (421 to 460 s), but LH2 is not very dense and needs a heavy tank and extremely cold temperatures. In contrast, rocket-grade kerosene (RP-1) can also be burned with LOX. Specific thrust is lower (300 to 340 s) but kerosene can be stored at room temperature and is denser (so needs less weight for its tank). See [116] which discusses the engineering tradeoffs. One of these is the smoothness of the ignition process for kerosene where chemicals must be used; this is perhaps another target area for MEMS. Methane can substitute for kerosene and, burned with LOX, has a specific impulse of 380 s. Though methane is less dense, it ignites more evenly and so can be used with spark-based ignition.

In 2010 Silvera and Cole [118] studied the potential impact of metallic hydrogen on space-flight. This substance is believed to be constructable but is not yet in existence; it is estimated to have a specific impulse of 1700 s.

A different approach uses hypergolic (self-igniting) fuel combinations such as hydrazine with dinitrogen tetroxide NTO (). The study [116] notes various niches where such fuels are appropriate but also cites toxicity concerns. Many other approaches exist. For instance, one can use a liquid oxidizer but solid fuel, a hybrid design [14], as in Virgin Galactic’s Spaceship III [121]. And other exotic fuels include liquid fluorine replacing LOX.

While many combinations of fuels are used for appropriate niches, calculations change if fuels, too dangerous for current use, become feasible via systems that incorporate MEMS, electro-optic sensors, and rapid computation to prevent adverse events. Either with separate units or with a dense internal network, such systems would exert tight control of the local environment.

However, there are other, non-rocket-based, concepts for getting into earth-orbit and beyond [78], at least theoretically. (We discuss electric propulsion methods below.) One extreme idea is an elevator from the earth to a geosynchronous satellite; the relevant Wikipedia page [95] traces its history back to the 19th century. Recently, another proposed version posits a space elevator anchored on the moon, reaching down to near the height of geosynchronous orbit [105, 106]. Unlike the original idea, which required materials not yet achievable, Penoyre & Sanford claim their proposal is feasible today.

An alternative means is closer to Verne’s vision - namely, a mass-driver. This is also considered by O’Neill [100].

According to a proposed scheme for maglev travel [25], one could travel at a top speed of about 5,000 miles/hr, a bit over 2,200 m/s, in a frictionless monorail operating in a near-vacuum, neutral buoyancy tube suspended just below the ocean’s surface, making the New York to London trip in an hour ([26]). At 1 G, one gains about 10 m/s each second and a simple calculation shows that it would take under 20 minutes to reach the 25,000 mph (i.e., 11,200 meters/sec) escape velocity from earth. A door at the top opens fast as the vehicle exits the tube and lasers in the vehicle vaporize just a few centimeters of the atmosphere directly ahead so that there is still no (or very little) air resistance. I thought this last notion might be beyond current reach, but just discovered that the concept is not only known but has been demonstrated to reduce drag by 50 percent [31]! More extreme notions have been patented by U.S. Navy Scientist, S. C. Pais, but the verdict is out on their feasibility [18].

Of course, it would be inconvenient to have a 4,000-mile-long ramp but a tube below the Pacific ocean might work.

Also, power could be supplied exogenously to a vehicle leaving earth or already in space, using solar reflectors or lasers, providing sufficient energy to enable electric propulsion (either ion drive or electrostatic). The European Space Agency (ESA) defines electric propulsion as “use of electrical power to accelerate a propellant by different possible electrical and/or magnetic means” [32] and they list about a dozen different types. In contrast, NASA includes solar electric power (SEP) [85] which has solar arrays that unfurl after launch, providing power enough to eject ionized xenon gas at about 30 km/s (65,000 mph) using a new design for Hall thrusters with magnetic shielding.

The solar arrays currently being tested open like fans or window shades [85]. The details are obviously not yet revealed but perhaps MEMS, combined with a novel, mathematical design [44], should also be studied.

For reviews of electric propulsion, see [48], on electrothermal, electromagnetic, and electrostatic types, cf. [65] on smart nanomaterials. Such propulsion systems skip the extra mass of rocket fuel and big engines, making reuse of transport vehicles much easier. The same idea of electric propulsion works from earth orbit to the moon (and return) and for deep space missions.

Non-rocket [78] propulsion applies especially to lunar launches. And it might be possible to use light pressure, either from the sun or from an orbital chemical laser to push a sail deployed by a spacecraft. The pressure of light on a square meter (at earth’s distance from the sun) is about a miligram (i.e., 10 micronewtons of force). But a 10-square kilometer sail, if permitted by material science, would receive 1000 Newtons of force from the sun, and this might be supplemented by a large solar mirror-array or laser.

In planning, it is wise to consider the possibility of new technologies. When the author was at Bell Telephone Laboratories (circa 1980), the key technology competition was between analogue and digital systems. In the long run, digital has won out in most domains - though audiophiles may prefer the former. But at each step, the choice was not always easy, as the older technology kept improving while the newer technology often had unforeseen flaws. Also, technologies are interconnected. A reasonable criterion is to have superiority of at least an order of magnitude (i.e., factor of ten) in the new system before replacing the old.

5 Economics

As Robert Heinlein put it, “Space is the high ground.” Every major world power needs a space program for national defense. I believe the NASA projections are much too conservative but that their emphasis on basic science is correct. A very strong economic reason to go into space is for planetary safety.

Orbital satellites might be used to defend earth from in-coming asteroids but to do so will take enormous power, meaning solar collectors or nuclear or chemical power plants to run the lasers, and at present, building these can’t be done economically if construction materials have to be lifted from earth into orbit. Space utilization for planetary defense needs a lunar base.

NASA lists various benefits for the International Space Station (ISS) [83]:

  • Economic valuation

  • Scientific valuation

  • Economic development of space

  • Innovative technology

  • Human health

  • Earth observation and disaster response

  • Global education

I will summarize their website. There is economic value in studies already conducted on the ISS. For instance, ordinarily, fire burns at 1400 F (760 C) but a series of experiments (FLEX), aimed at fire suppression and soot production in micro-gravity, have discovered so-called “cool” flames at 370 C and 200 C. Other promising topics are earth observation, microfluidics, robotic surgery, and protein crystal growth, all of which are already involving companies working with NASA, and the ultimate value of fundamental geological, atmospheric, oceanographic, physical, and chemical knowledge is enormous.

They point out that since 2006, pay-for-performance (rather than cost-plus) for ISS has catalyzed venture capital to support space industry start-ups. Much of the commercial activity has been in Low Earth Orbit (LEO) and launching of satellites for telecom and terrestrial observation.

Most of the ISS’s value is yet to come but just the data on the astronauts themselves, as the first people to live in space, is already vital. The full NASA study [84, p. 45 ] found “the first evidence ever that improving nutrition and resistive exercise during spaceflight can mitigate the expected bone mineral density deficits historically seen after long-duration microgravity missions.”

The ISS orbits at 254 miles, while the earth’s magnetic field extends to (about) 40,000 miles (65,000 km). Thus, the ISS is protected from cosmic rays and solar-flare particles compared to Luna which is about 239,000 miles (384,000 km) away. Geostationary orbit is at 22,000 miles (36,000 km), so probably radiation there will be higher than on the ISS but less than on the moon. Experiments on ISS [84, p. 47] have shown that kevlar gives both impact-resistance and radiation protection, leading to theoretical upper bounds on the cost of space architecture to shield its occupants from danger.

The economic incentives to get out to deep space are huge. Energy from the sun will be readily available even with current not-too-efficient technology as the direct solar flux is so large and there are Lagrange points where giant mirrors will never be in the shade. Micromechanical systems (discussed below) could allow mirror arrays to focus their energy to produce power in continuous mode or, in pulsed mode, to destroy small meteors.

Space and lunar manufacture draw on virtually unlimited material resources and create no pollution of air or water. Low (or no) gravity can reduce costs and may permit types of manufacture and medical services that would be impossible under ordinary conditions. There are asteroids, and moons of other planets, and most have even lower escape velocity than the moon. The solar gravity field makes slow transport to earth-moon inexpensive.

6 Architectural

A research group from the European Space Agency, MIT, and Skidwell, Owings & Merrill architectural firm have a plan to build such a base using materials available on the moon [58]. They point out that in its pulverized form, the lunar regolith is well-suited for various fabrication processes and contains most of what’s needed (except nitrogen).

This idea, of mining the moon for useful substances, and taking advantage of the much lower lunar gravity well, was previously suggested by O’Neill [100] for an L5 space colony, and the same logic applies to all space construction.

The tricky part is to come up with a specific plan for just exactly how to do it. We sketch some ideas below.

Two-tiered approach

For Lunaport itself, we think that a two-tiered approach is going to be needed: an outermost macrodome and, within it, smaller structures, some of which are mobile. The macrodome covers the entire area of Lunaport and protects internal structures from UV and other radiation as well as micrometeoroids. For definiteness, we take a radius of 2 km for the disk-shaped region covered; the height of the dome will be determined by engineering constraints (chiefly the materials used), utility of the resulting structure, and cost.

A key parameter is the largest size meteoroid that the macrodome will be able to stop. The smallest meteoroids, under about one gram, are called micrometeoroids, and they may be stopped by a so-called Whipple shield, proposed by Fred Whipple in 1947. (It certainly constitutes one of those “good simple ideas” that Efron wrote about.) A “sacrificial” outermost layer is separated from an inner layer by a small gap so that a micrometeoroid striking the outer layer produces a plasma shock-wave but the separation is sufficient to attenuate the force of the shock-wave sufficiently that the inner layer can withstand it. The outer layer can then be replaced. However, the shield was designed for spacecraft, where peak velocities of micrometeoroids are in the 3 to 18 km/sec range [130]. On the moon, speeds can be four times higher. A paper from 2015 [75] models the Whipple shield on the impact of a 1.4 mm aluminum sphere on a 0.6 mm aluminum plate at 9 km/s but it is not clear that this applies to the lunar surface. Recall that 30 grams is about 1 ounce and 1 km/sec is about 2,200 miles/hr. Now imagine stopping a one-ounce object traveling at 40,000 miles/hr! See [76] for more on shields.

The surface area of the moon covered by Lunaport (under our 2 km radius assumption) is roughly 12.5 square km. If there are 100 times more meteoroids in the 10 to 1000 gram mass range than occur above one kilogram, then based on previous estimates, Lunaport’s chance of being hit in a given year is .0125. Thus, meteoroid shielding is vital.

I wonder whether some very large web-like array of a gooey metamaterial, extending above the macrodome even by several kilometers, could somehow stop an arriving meteoroid and even manage to collect some of its incredible kinetic energy. Given a sufficiently thick shield (easily re-extruded if hit) built with cross-linked carbon fibers, this might be able to stop everything under a few kilograms. But if something really large were coming in, the only hope would be defensive lasers or solar reflector arrays (see below); luckily, larger objects are more easily detected even when far away.

Under the macrodome

The inner domes (or kiosks) will provide air at ordinary pressure as well as heat and, when necessary, light. If certain types of radiation shielding are sufficiently expensive, then it would be restricted to the kiosks. These kiosks would be purpose-driven, e.g., arrival/departure sites, health, administration, supplies, shopping, restaurants, hotels, and recreation parks. To travel between fixed kiosks, people would use provided individual or family/group vehicles which are mobile kiosks. Administrative matters could then be handled via telescreen from the vehicle.

No “driver” would be needed as all vehicles will read an RFI tag worn by every person in Lunaport, whether passenger, crewmember, or support personnel including security. Such vehicles will only be operable via the tag-link giving suitable permissions. Additional biometric safety/security protocols may be required to reach a higher level of authorization.

Passengers might be free to choose where they wish to go in the meanwhile as their departure time and place will be known by their transporting vehicle. The low-G environment (about 1/6th Earth’s) could afford interesting opportunities for development of tourist attractions such as trampolines and rock/wall-climbing.

Traditionally, cities have evolved around ports, and the city of Selene might indeed grow up in close proximity to Lunaport, perhaps even below it as the bedrock of the Moon will provide protection from meteors and radiation.

Emergency preparedness

Dealing with emergencies is clearly of primary concern for a station on Luna but the presence of gravity offers the possibility of an architectural and mechanical design fail-safe to protect humans from the vacuum. After a major explosion (whatever the cause - terrorism, meteor-strike, industrial accident), all electrical power may be out - except that given by batteries and auxiliary generators. The communications system would likely be down. Absent Star-Trek’s Capt. Kirk or some other improbably timely hero, what could be done to protect the travelers if such a catastrophic event should happen?

One solution might be to release gravity-powered weighted “curtains” (like the water-tight doors on a submarine) to protect the habitable areas. This failsafe could be initiated manually by using a battery-operated laser, taken from a wall-mounted break-glass case, to trigger a small explosive release or it could be set off automatically by the loss of ambient power or air pressure.

The curtains would need to descend quite rapidly; even fractions of a minute could be devastating - as has been vividly simulated in film. One can imagine systems capable of doing this job if granted nanotech (to make it happen extremely fast) and AI (to avoid hurting or killing people who are in the way). Currently, neither of those two scientific domains is near the required level.

If some of the kiosks are sufficiently large, the ideas mentioned above (such as protective curtains or doors) would be applicable. The area outside the kiosks is not required to be habitable and so could be retained at pressure high enough to prevent lunar dust from infiltrating the macrodome but low enough to minimize the internal windstorm caused by an explosive breech.

Such detailed consideration may seem out-of-place in a “high-tech” study but we think it is important. For instance, Dr. Robert Ledley, recipient of the (U.S.) 1997 National Medal of Technology and Innovation for his invention of whole-body CT-scanners emphasized how he provided a counter-weight so that neither patient nor physician/technician could be hurt mechanically by the large and heavy apparatus involved [61].

The Shackleton Crater project

In an interesting article [73] from 2008, Edward McCullough studied a very ambitious design: a 25-mile diameter, mile-high dome over the Shackleton crater, near the Lunar South Pole. Currently, SpaceX seems to also be considering this as a location for a base [40].

McCullough proposed a structure built with hexagonal or pentagonal glass patches. This is consistent with a planar map model of the dome, where the patches correspond to the “countries” of the map. By the famous 4-color map theorem [111], it is possible to assign the patches to four distinct types in such a way that any two patches which share a common side are of distinct types. Further, if all patches have at least 5 sides, then there must be at least 12 pentagonal patches if the patches cover a sphere, and again for a sphere, if no patch has seven or more sides, we should find exactly six pentagonal patches, the rest being hexagons. McCullough’s design [73] uses a titanium framework, built by robots, to hold the patches in place, and the structure is to be anchored to bedrock. One can also arrange that, at each node of the framework, exactly three struts meet.

An alternative “dual” approach [111, p 180] is a triangulated planar surface such as the geodesic domes described by Buckminster Fuller [38]. Here the glass panes are triangular and nodes are incident to five or six struts. Nodes can be partitioned into four types so that any two nodes which are endpoints of the same strut are of distinct types. Thus, the duality interchanges patches and nodes analogously to countries with common borders vs. capitol cities connected by rail-lines (or highways) crossing the common boundary.

Which is the better model for a protective dome? We think this is an interesting question from the standpoint of mechanical engineering.

A very well thought-through plan has to include the provision of materials to build the outermost dome, including the robots to build it!

So before building Lunaport, we may need to establish a robotic factory on the lunar surface which can transform the pulverized regolith (lunar “soil”) into material for robot construction and for the construction of struts, and polygonal 2-dimensional facets to build the outer dome.

McCullough proposes glass patches be built in layers to control heat stresses; in his model patches are 2 to 3 meters thick and are equipped with both maintenance and catastrophic repair mechanisms. He observes that the regolith contains minerals in finely pulverized form, perfect for fabrication, and that there is a substantial amount of oxygen present in the regolith, but that nitrogen may be more difficult to acquire [73]. However, coverage of “damage control” seems rather thin, and the article does not discuss radiation hazards; it also gives no account of the danger from meteoroids.

Constructing Lunaport

It is worth wondering how the dome would be erected. One naive approach would be to have some robotic spiders build up a heuristic geodesic dome, so that junctions and struts (fitting together like a “lego” structure) form a triangulation of a hemisphere or a structurally optimal curved surface. As the outer ring, where construction would begin, will not be completely level (and heights could change a bit due to the weight of the dome), it would be helpful if the last closing struts are of adaptable lengths.

Once struts are in place, each of the resulting triangles is filled by a pane of “glass,” where we use the word to mean merely some transparent and strong material with proper shielding characteristics. (But transparency might be negotiable! See below.) The robotic spiders would carry a fabricated pane and use it to fill in a triangular (or polygonal) facet. It would be best to include MEMS elements in the struts to enable them to adapt to slight irregularities.

Given a 2-km radius circular dome, the ground coverage is square kilometers (about 5 square miles). Estimated frequency of meteor strikes suggests that occasional damage will occur both to panes and struts, and the robots will need to be able to repair it. Thus, structures need to be modular and constructable heuristically, rather than following a strict algorithm with geometric regularity. This is another topic needing both mathematical and mechanical insight, including the ability to construct the correct patch.

Maintaining a slight internal pressure within the macrodome will push the panes against the struts, thereby making the interior inaccessible to the destructive dust of the lunar surface. Entering vehicles would need to be cleaned off thoroughly. In the event of a decompression event, a backup mechanical linkage will be needed to ensure that the panes don’t fall out.

But before construction, one must know what one is building. Careful study of radiation, dust, and meteor risks, and how they can be ameliorated by structural features is key. E.g., is 10-cm-thick or 1-m-thick shielding needed?

In addition to shielding, buildings on the moon might be protected from some meteors by using high-powered lasers or reflected sunlight to destroy the hurtling chunk of rock. If the meteoroid is traveling at 50 km/s, there might be at most two seconds of exposure, so vaporization needs a tight focal spot to be maintained. If the rock is large, it might be destroyed by pulsing the lasers, or reflected sunlight, acousto-optically in order to set up a destructive vibrational resonance. Such a “Tesla ray” would apply to mining the asteroids, and if it worked at small scales (and commensurately high frequencies), then virus particles might be selectively destroyed.

Traveling within Lunaport

There are two types of vehicles to consider, those staying within the macrodome and those that venture out. The latter will need to have shielding (if they carry humans), and even robotic vehicles will need ruggedized electronics to cope with the solar wind and cosmic rays.

Both types of vehicle need to have potable water, snacks, and rudimentary sanitary arrangements for waste disposal but in emergency situations, larger vehicles should be available to pick up a smaller vehicle and put it into a protective kiosk - e.g., if there is a failure of the air supply. Another purpose of the passenger vehicles is to provide a comfortable and private place to travel about and see the sights.

Lying on your back, looking upward into space, through the roof of your vehicle toward the covering macrodome, what will you see?

Attention should be paid to human psychological comfort. Though experimentation and experience will be required, at a guess, the intensity of the sun should be very much reduced (so that the viewer is dealing with a light-source comparable to the sun at earth’s equator), while the stars should be displayed at one or two orders of magnitude beyond what we see on earth, all appearing in the black sky. Both would be projected onto the macrodome by lasers in simulation of the actual sun, stars, and planets in the sky; the earth’s image would be projected by laser video. Thus, the organically tangled shielding above the dome, which blocks and absorbs radiation and meteors is invisible from inside the dome. A vehicle trip outside the dome will show them as they are if a passenger is willing to risk the absence of meteor shielding.

In the event that the macrodome alone is sufficient shielding, if it diminishes sunlight sufficiently for human comfort, the stars would likely be invisible. So psychophysically speaking, video projection may be preferable to create the feeling of “being in outer space.”

As for buildings (i.e., kiosks), while the outer aspect may be domed, the inner portions might still consist of rooms including some with non-square walls, floors, and ceilings. Perhaps the psychological advantages of a familiar rectilinear architecture will outweigh any design and operational awkwardness.

There might be a need for environments that don’t feel like they are in an airless void - such as a pseudo-Miami with pools, sun, and blue sky or a midnight sky in the far North. If the trip becomes easy enough, specialized sports for the 1/6-G gravity might arise. Using video, crowds could watch from earth as players and teams compete in these lunar games.

Mechanical engineering for transport networks

In supplying minerals to build the macrodome and for manufacturing plants, the simplest solution might be the old idea of a railroad. Starting with a blank slate (and so no need to be compatible with previous equipment), I envision a few key properties: Laying track should be easy, inexpensive, and reconfigurable - e.g., as the water-ice is mined from within the sheltered craters or the now-known tunnels. Vehicles don’t need to travel fast but they do need to use MEMS to ensure reliable performance so small deviations don’t cause derailments. Track has to be modular so it can be replaced in the advent of meteorite hits or lunar quakes.

The vacuum environment must be considered in planning the mechanical engineering aspects of construction. For instance, electrodischarge machining (EDM) uses a dielectric fluid. A target for metamaterials would then be to create a “nano-dust” with dielectric properties. This wouldn’t work well on earth because of wind and rain but it should be feasible on the moon if the metamaterial can be made. A recent article (Lipiec et al.[67]) describes advantages and disadvantages of EDM in drilling. Perhaps the metamaterial approach can shift the balance favorably. In advanced planning for lunar manufacture, it will be important to also pay close attention to the impact of process parameters on the quality of machining, Machno [101].

Designing lunar transport systems for goods will require the integration of mechanical, electronics, and complex information processing (now called mechatronics); see, e.g., Fijalkowski[41], Schoener [115]. If inexpensive modular track-laying can also be used on earth, the result could be good for freight transport.

7 Enabling technologies

Often, it is the little things that define a radically new environment. In addition to robotics, our next section, there are a variety of supporting systems on which everything else depends. To build Lunaport and to have people use it safely requires small-fast-smart devices, powerful-and-accurate computation, materials with unusual properties, high-bandwidth secure communications, and intelligent systems. We take these five items in turn.

Microelectromechanical systems (MEMS) vs. nanotech

Microelectromechanical systems (MEMS) consist of mechanically active elements operating at the 1-to-100 micron scale under electronic control and integrated into a chip, and they include both sensors and actuators. MEMS is a mature but still expanding technology with very favorable properties including low unit cost (when mass produced), high sensitivity, very fast activation and repetition, and very low power consumption.

MEMS was invented in the 1960s by H. C. Nathanson for application to radio tuning, but he obtained a further patent in the early 1970s for micro-mirror arrays in display. In recent decades, mechanical capabilities of the microscopic systems have become stronger; see an article from 2008 in MIT’s Technology Review magazine [96], where MEMS-based valves operate faster, allowing air conditioners to function for 25 percent less cost.

This technology is already used in the dedicated system that auto-adjusts screen orientation on a smartphone [92]. More generally, MEMS are used in accelerometers (and so in vehicle airbags), gyroscopes, digital image projectors, miniature pumps for medical devices, and inkjet printers. A discussion of advantages and disadvantages of MEMS in the context of radio frequency applications in telecom is given in [93]; see also [94].

For space and lunar applications, MEMS will have particular advantages. A known deployment problem for MEMS [91] is packaging; components (electrical, optical, and mechanical) must be protected from air and water. This won’t be a problem in space and lunar environments, though in the latter one may need to account for the dust.

Another advantage (for space or the lunar environment) is that the electrostatic forces which operate many of the micromachines in MEMS [99] are perfectly suited to the bone-dry vacuum of space. In fact, microelectromechanical systems can be used for propulsion in space and indeed electric propulsion systems using an electrostatic (gridded ion) field or a combinations of electric and magnetic fields (Hall effect thruster) have been used there for some time, see, e.g., [108].

Perhaps electrostatic systems might be efficient propulsion for vehicles within the macrodome of Lunaport. If it is possible to maintain a thin atmosphere within the outer dome, one might also be able to design drones - providing security, sensing, and supply.

Nanotechnology consists of those devices and structures in the 1-to-100 nanometer scale. This is still largely theoretical, except for some applications to material science by creating substances with “enhanced properties such as higher strength, lighter weight, increased control of light spectrum, and greater chemical reactivity than their larger-scale counterparts” [128].

The article [72] lists several current uses of nanotech: sunscreen has nanoparticles of titanium dioxide and zinc oxide, clothing has nanoparticles of silicon to promote water-shedding and similarly for furniture fabric. By adding carbon nanofibers, flammability can be reduced by one third. Nanotech can improve adhesiveness in high-temperature environments, and it is used in tennis balls and racquets for improved strength.

Nanotech is fundamental to the fabrication of chips, where feature size is now below 10 nm. Furber [35] discusses the evolution of chip technology in the last 50 years: number of transistors on a chip has gone from to , bits on a chip also from to while transistor feature size has dropped from 10 m to 10 nm, which means the area ratio is again a million-fold better. This is roughly 20 doublings of power in 50 years. However, according to Waldrop [129], the limiting size of features is 2 to 3 nm which corresponds to only a few atoms, and he and others predict the end of Moore’s law of exponential growth of power for integrated circuits. Thus, it seems that nanotech works for sensors and almost works for computing elements. Nanoscale movement is being studied [8] in biomedical applications.

In fact, there are alternative scenarios for computing elements, including dealing with electron spin (rather than movement), direct optical implementation of neural networks, where atoms are the processing elements, and quantum computation. These could replace chip technology.

However, in vacuum, another design would be possible; a faceted sphere, whose facets are inward-facing chips, communicating by laser and constituting a fully interconnected parallel computer. Thus, the next stage of digital evolution might not be in the individual chips but rather in how they are organized into larger units, just as in biology, the same basic cell-types appear in all mamalian brains [102], though the genes expressed (i.e., the programming of the cell) reflect the particular species [141].

Computation

The first type of computation is conceptually the simplest but often very difficult to do: choose among a finite set of alternatives to provide the best solution to a problem.

For instance, in Manhattan, New York, for much of the “mid-town” area, streets are in a rectangular grid, oriented North-South and East-West. Among traffic-light synchronization strategies, it is possible to aim for optimal N-S or optimal E-W but not both. As the area is much longer than it is wide, most traffic is N-S so indeed that is the chosen direction for synchronization.

If only one parameter matters (e.g., average travel speed for vehicles), the alternatives are naturally ordered by calculating this parameter; choose the one which makes it biggest. However, if two or more parameters are important but in different ways that are not easily related (e.g., safety and fuel consumption), decisions become much more difficult.

In fact, if more than two alternatives are considered and one has a group of decision-makers, then no “reasonable” procedure exists for a fair choice! This is called Arrow’s Theorem (for the economist K. Arrow), and the term “reasonable” means satisfying a few conditions such as an alternative, which is at the bottom of all decision-makers’ lists, cannot be chosen (e.g., [30]).

Deciding dichotomies of location will involve judgements that really can’t be quantified exactly - for instance, whether or not to build Lunaport at all! Discrete choices arise at all dimensional levels, e.g., which one of a set of candidate surface materials will be used for the dome?, so we now discuss various methods for selecting among a finite set of alternatives.

Means for making such choices do exist. One of them was proposed (and used) by Benjamin Franklin [10]. Over a few days time, the Decision Maker puts down a list of pros and cons and then trades-off roughly equal combinations of pros and cons, not necessarily on a one-for-one basis - for instance, removing the first two pros and the first three cons from the list - until it is clear which alternative is dominant.

Another method, that has the benefit of being usable by a group and that takes advantage of modern computational theory and power, is the analytic hierarchy process developed by Thomas Saaty [109] in the mid-1970s. An early use of the AHP was to allocate a major investment by the Kuwaiti government in the transportation infrastructure of Sudan, splitting the money among projects involving harbors, airports, roads, and rail under a set of possible future scenarios for Sudan’s national evolution [110], prioritizing the projects numerically using input from the decision makers.

The AHP first requires creation of a tree-like hierarchy. For instance, a young person, making plans for the future along with parents or friends, might put at the top, school, a job, or travel. Then for each of these, a subhierarchy - say a list of five schools or jobs or even a list of a few types/levels of school or work, and so on. Moving up invokes higher levels of strategy (e.g., ranking the criteria being used to choose between school and work - say, short-term economics vs. long-term career goals), while moving down drills deeper into tactical choices. Thus, users of the AHP build their own “expert system” style of software based on the algebraically justified calculation of specific vectors given input from the users.

For each set of choices within the hierarchy, an matrix is constructed whose entries are filled as follows: In the -th row, -th column, the DM or group of DMs choose a number that denotes the degree to which choice is preferred over . The rule is that if it is absolutely preferred, you use a , if they are equally preferred you use a , while if is absolutely preferred to , a is used. In situations where the preference is not absolute, is taken to be either or their reciprocals, and we enforce the conditions that and . From the matrix, a list of priorities is derived.

Humans are inconsistent so the numbers in aren’t exactly ratios of preference (i.e., is not the product of and

) but when the matrix is filled in by knowledgeable people, the resulting vector of relative preferences has been found to be satisfactory by the DMs. Saaty’s method for extracting the vector is quite similar to the now well-known Page Rank algorithm used by Google. The dominant eigenvalue for the matrix has a unique eigenvector which, normalized, is the required vector.

A benefit of Saaty’s method is the better grasp obtained by structuring the hierarchy but a drawback of the AHP is the large number of matrices that may arise. Further, the number of choices which must be made for each matrix grows quadratically with matrix size. If a set of 6 items are to be prioritized, the DMs must consider 15 distinct choices for that one matrix, while choosing among 15 items would mean 105 elements of the matrix to be chosen! Also, consistency of choice decreases as matrix size increases. Thus, the hierarchy can’t be too broad. But if the hierarchy is too deep, then minor errors may be compounded, and a deeper hierarchy means more matrices.

In fact, a logical, step-by-step process doesn’t seem to be the way that a really skillful executive (or an artist!) navigates among complex choices. We will propose a geometric approach to this later.

Sometimes, increasing complexity makes things easier. Large dimension is regarded as a “curse” (in Bellman’s famous metaphor), but high dimension can also be a blessing [49]. Many systems work best when they are near but not above capacity. Busy travel facilities are more efficient because more travelers are served by the same sunk cost in buildings, resource requirements (e.g., staffing) remain predictable, and economy of scale applies.

When a data-structure is very large, even simple queries can be too slow. For instance, finding an endpoint in a linked list representing a random tree (a connected graph with no cycles), on labeled vertices, can be done either by running down the list until you come to a vertex with a unique neighbor or alternatively by following a path in the tree until (as the tree is finite) you reach an endpoint. These step-by-step procedures (algorithms) require a number of operations which grows with the logarithm of .

In contrast, Renyi proved that the heuristic which chooses at random an element of the list of vertices and checks to see if that element is an endpoint has a chance of about of success as gets large. But this asymptotic convergence is very fast! If and the tree is chosen, uniformly at random, from the set of all trees on 10 given vertices, the value is already within three decimal places of agreement with the limiting value. The chance of 20 wrong guesses is about independent of n. The guessing heuristic can be parceled out to multiple processors since no coordination of results is required. See [50, pp 282–283].

Heuristics are often based on broad features of a problem and can succeed even in situations where assumptions don’t hold. Applying the end-point-finding heuristic to a large graph where a few extra edges have been added to join vertex-pairs in a tree (i.e., where the graph does have cycles but only a few), the same heuristic works with slightly diminished probability, while the path-finding algorithm will fail if one is walking along the cycle.

Two good simple ideas are (1) be greedy and (2) do the harder parts first. Not only do these two heuristic work very well in the context of putting weights into “bins” (called bin-packing), but one can prove it! See [49, pp 289–290]. The combination of (1) and (2) is called First Fit Decreasing (FFD): Given weights between 0 and 1 and unit-capacity bins, reorder the in decreasing order; then place each weight into the first bin into which it fits.

There must exist an optimal way to assign weights to bins (as the sets involved are finite), but practically speaking this optimum is unavailable (curse of dimension) because, as increases, a “combinatorial explosion” makes the number of possible assignments too large. Using FFD, one is guaranteed to use at most approximately 22 percent above the optimum but (blessing of dimension) with a large number of weights selected at random (within the unit constraint), the expected performance of FFD is within one bin of optimal.

Many large-scale computations now use neural networks. While some of the mathematical theory has been worked out (e.g., [54, 55]

), the good behavior of deep networks for machine learning (in interesting niche applications) is not well-understood, nor are the limitations of this technology.

See also evolutionary computing (e.g.,

[77] has a list of standard texts in slide 3); for theory, see [68, p. 276]) on the Banach contraction theorem.

We continue the discussion of computation in the mathematics section.

Material science and metamaterials

According to the Center for Metamaterials and Integrated Plasmonics of Duke University [17],

Metamaterials are artificially structured materials used to control and manipulate light, sound, and many other physical phenomena. The properties of metamaterials are derived both from the inherent properties of their constituent materials, as well as from the geometrical arrangement of those materials.

Of course, the “artificial structures” involve numerical parameters derived by mathematical methods and computational techniques. For instance, a review by Steinhauser & Hiermaier [120] discusses the simulation of molecular dynamics and density functional theory for electrons in applications to polymer physics and shockwaves for medical devices related to lithotripsy. Machine learning and an innovative use of Principle Components Analysis has facilitated band gap maximization for an acoustic metamaterial (Gnecco et al. [44]). These new materials can have useful properties [117], [133], [74].

An interesting possibility to produce materials with great strength but low weight is to use a fractal-like construction to remove material. For example, consider a 3-by-3 cube of cubes, which consists of 27 small cubes comprising the solid object. If we remove the middle cube from each of the large cube’s six faces, the resulting object might have equal strength to the full cube while it has only 21/27 = 7/9 of the mass. If a similar transformation is applied to each of the small cubes, the mass is now down to .

If this could be done 10 times to a 1-meter cube, corresponding to a linear dimension of about 16 microns, the resulting metamaterial would have only 8 percent of the mass. If such a removal method can be found which does not affect the strength with respect to forces (perhaps only in some directions), then the metamaterial would be very useful. Such fabrication might be done by nanobots; that is, autonomous nanotechnology robots. Another route could be a 3-dimensional printer of the sort used to fabricate airplane parts (see Bellamy [3] and Salmi [4]). Thus, additive manufacture might have advantages for such metamaterials.

Lasers, photonics & electro-optics

Subtractive manufacture, on the other hand, can benefit from high-power laser beams to ablate a substance in very precise ways, just as ocular surgery is now often done by laser.

Laser technology is used in aeospace construction in various ways [5], as well as in communications, power transmission, mining, and basic science - in addition to visual displays, material inspection, and last but not least, health and medicine. For the latter, argon laser treatment (488 and 514 nm) reduces gamma ray damage [6] while 650 nm (red light) from a semiconductor laser protects wheat against UV-B [7]. In fact, red and infrared lasers and laser diodes are now recommended for prophylactic treatment of head and neck cancer patients prior to radiation and chemotherapy [142].

Other uses of photonics include sensing, material processing by light and heat, and electro-optic and acousto-optic scanning in mirror deflection. A significant recent development is the fabrication, via conventional CMOS chip technology, of photonic MEMS switches for datacom by S. Han et al. [39]. Another application of conventional CMOS/MEMS process has made possible small and inexpensive ion motors [113], intended for use in space.

An even more revolutionary result is in the just-published paper in Nature Communications by Buddhiraju et al. [11]. This group, from Ginzton Laboratory, Department of Electrical Engineering, Stanford University, have built a photonic architecture that can achieve arbitrarylinear transformations. They claim that their approach is compact, scalable, and reconfigurable, and that it works in both quantum and classical domains. Their approach involves very fine control of frequency. According to the paper, there are “wide-ranging applications in frequency metrology, spectroscopy, communication networks, classical signal processing and linear optical quantum information processing.”

Artificial Intelligence

In a 2019 lecture in Chicago [63], Douglas Lenat (inventer of the LISP language) asserted that

“There are 2 different ways to ‘power’ an AI,

  • Statistics (induction, machine learning)

  • Logic (deduction, abduction, causal models)”

He omitted a number of other potential sources of power for an AI, including mathematical analogy, physical forces, and the critical ingredients of intuition and desire.

Just as airplanes fly in a very non-bird-like fashion, it is hoped that one day machines will “think” in a useful (if not human-like) fashion. But as there is no clear definition of what thinking actually is, the task of having a machine replicate or even improve on human thought is ambiguous. Aside from lacking “common sense” [64], machines can only learn what they are programmed to learn, and the adaptability of most AI is low. Thus, the cost of having a very expensive team encode a particular human task so it can be done by machine has to be repeated each time the task changes by a small amount - which is an unavoidable fact of life in a dynamic and rapidly changing technological world.

Other serious problems with artificial intelligence include the ELSIs (ethical, legal, and social issues) and also the black-box nature of the beast, which does what it does but doesn’t always give reasons. The recent success of deep neural networks and machine learning in Google Deepmind’s victory over a human Go master involved teams of very talented humans in the domain areas as well as in computer science.

Though deep nets have done well in certain niches, would you trust your grandchildren’s safety to a system based only on deep networks? In my opinion, automation of jobs on earth should only be used to increase the efficiency of the work and the satisfaction of the worker, while artificial intelligence and machine learning should only be deployed when they substantially enhance both safety and the perception of safety.

Recently, a very interesting idea has been advanced by Gorban and Tyukin [36] to apply an abstract mathematical idea, “concentration of dimension,” discussed in section 9, to the correction of legacy AI code. Their idea is to override the pre-existing code in certain situations based on special geometric properties of random point-sets in spaces of dimension much larger than the number of points, which enable one to find a separating plane for the situations to be corrected.

It follows from earlier work such as the Johnson-Lindenstrauss Lemma (JLL) [21] that linear projection from a -dimensional dataset of points down to variables preserves the geometry if isn’t too small compared to . Donoho and Tanner [24] rediscovered this fact and related it to a problem in the geometry of Platonic polytopes. They found (via simulation) that the thresholds are very sharp for this domain, where the dimension of the dataspace is much larger than the number of datapoints.

The JLL [21] says that for and for a set with points, if , then there is a map such that for all ,

with denoting Euclidean distance (in any Euclidean space). Such an preserves distances between points of up to a multiplicative factor that differs from 1 by at most . In fact, almost any linear projection will do and the random projection matrix can be further constrained [60]; cf. [52, 53, 36].

8 Robotics

Robots are not light-weight machines. According to one article [104] “A robot with a lifting capacity between 5 and 7 kg has about 300 kg average weight.” Simple Newtonian physics suggests that to move a massive object quickly and smoothly requires another massive object. Thus, Lunaport’s construction-bots (and the macrodome itself) must have a large number of small units.

In fact, there is a specialized area of AI which gives an organizing principle for such an approach: swarm intelligence. One website [123] describes it as

an emerging field of biologically-inspired artificial intelligence based on the behavioral models of social insects such as ants, bees, wasps, termites etc.

Several aspects are mentioned: agents (that is, the individual units) choose actions (one assumes from a list of innate behaviors) and carry them out. Cooperation among the agents creates emergent behaviors which solve problems. In weak emergence, behavior of agents can be linked to something specific while in strong emergence, the agent’s behavior appears to come out of the blue. A critical distinction of swarm intelligence is that agents interact by modifying their common environment; this is called stigmergy and makes context-awareness the channel by which the agents interact. Positive feedback occurs by having more agents adopt successful emergent behaviors, while negative feedback arises by preventing mutual entrainment (i.e., convergence to a single behavior). Such negative feedback is achieved by randomness and social interactions among the agents. Optimal functioning is ensured by a proper balance in positive and negative feedback, and [123] claims that this is indeed the case in nature. The three chief swarm behaviors are foraging, nest-building, and moving in the environment.

A swarms model for Lunaport’s construction

The sequence of tasks is as follows: (i) find suitable materials on (or near) the lunar surface (within the region where Lunaport is to be built), (ii) dig them out, (iii) transport the materials to a refinement site, (iv) refine them, (v) transport the refined materials to a fabrication site, (vi) fabricate struts and panes, (vii) transport them to the site of the macrodome, (viii) assemble the macrodome, (ix) repeat stages (i) through (vii) replacing struts and panes with the metamaterial shielding for radiation and impact described earlier, (x) apply the shielding to the outside of the macrodome, like whipped cream on a banana split - ideally using hoses attached to the outer surface of the dome.

Each stage could be associated with a swarm of bots. Of course, one bot might do several tasks but most of these seem to require special attributes that are not necessarily well-suited to the other tasks. Prospectors, miners, and trucks cover (i) through (iii). The refinement site itself would have to come pre-built, unless recursively the refinement facility can be handled in the same fashion. In that case, the above description applies first to the location of materials and construction from them of the refineries - prior to Lunaport.

Continuing, we need refinery workers, a different set of trucks for the refined material (if needed), fabrication (i.e., manufacture) workers, and so on. There is a chicken-vs-egg problem here, as the robots need to be built. Shipping from earth, under most reasonable scenarios, would be very expensive, so one could build the bots on the moon. But that involves much more complex assembly than the simple geometry of a “geodesic dome.”

The key will be how versatile the individual robots can be. If we could achieve the competence of an individual ant (which can do many different things and can lift several times its own weight), then it might be able to unify a number of the roles.

Soft computing and swarm intelligence (also called metaheuristics) were applied to modeling of transport problems by Lucić [69]. His thesis described (feedforward, artificial) neural networks. The figures [69, p. 161]

) suggest to me that a combination of ReLU and Heaviside neural networks could handle the computations quickly. In

[70], Lucić and his thesis advisor, Teodorović, used a swarm method to solve a difficult combinatorial problem.

What is a robot?

One website [1] lists 7 challenges in robotics: “Manufacturing procedures, Facilitating human-robot collaboration, Creating better power sources, Mapping environments, Minimizing privacy and security risks, Developing reliable artificial intelligence, Building multi-functional robots.”

The key is what one means by robot (in Czech, “worker”). A robot that replaces managers has to do all the various things done by managers, while if the robot is replacing the finishing workers who paint the outside of an assembled vehicle, then only that single function is required. What designs will be useful for the special-purpose bots we have described?

Manufacturing is simplified if the bots are small. Each bot would then consist of a functionally organized collection of MEMS, organized into a standard, rugged 3-dimensional shape, containing articulated linear segments - like legs and feet and toes and also slender extensions like antennae or whiskers. Perhaps the size of one or two fists. Effectors might be organized into opposing pairs, as are muscles in the body. A very desirable amplification process would be a cascade initiated in a network of MEMS-effectors to increase the force. If this can be done at nanoscale, it might provide an active meteor shield.

Human-robot collaboration doesn’t exist in this scenario. Power sources are easier for a smaller bot and with the solar energy available, this should be possible for lunar surface activities. Mapping the environment includes not just the prospectors but all bots that work in an environment with others of their kind, as well as with evolving structures. It is a potential issue for all the bots. Computational space needs to be allowed to have no-go zones, obstacles, with the non-obstacle space being no longer simply connected. Each bot understands its motions with respect to a mathematical environment which is built according to the bot’s observations and a family of prescribed rules. The swarm of bots uses stigmergy to organize their world.

Privacy and security risks seem lower priority, and building multifunctional robots is a different job. All we are trying for here is design for the lunar environment of specialized, self-actualizing tools. And the question of reliable artificial intelligence is mostly obviated by the predictability of the tasks. If it would be necessary to build the dome given some lunar crabs which could be found unexpectedly attached to struts, then the assembly bots would need to cope but (as far as we know) such events can’t happen. But flaws in existing elements can occur and coping behavior should be programmed into the bots.

9 Mathematics

Space travel is at the cutting edge of technology and so involves a large number of parameters; hence, one must consider the issue of dimension in computation which is required to deal with “big data.”

Dimension is a fundamental concept in algebra, topology, and analysis). An intuitive definition of the dimension of a Euclidean vector space is the number of pairwise-perpendicular directions, where two (nonzero) vectors are perpendicular if their dot product is 0. A basic theorem of topology (“invariance of domain”) states that -dimensional space is homeomorphic to -dimensional space if and only if , i.e., dimension is a topological invariant. But a data-set contained in -space might possibly be concentrated near an -dimensional submanifold with - e.g., a 1-dimensional curve - and so one could think of the data’s intrinsic dimension as 1 with the data scattered around the curve. Determining true data-dimension may be possible via topological data analysis [15].

High-dimensional data-spaces have unusual geometric properties [24], and many aspects of modern computational science are involved with high dimension. Even moderately many dimensions (in the dozens, rather than in the thousands) is enough. Digital coding for radio transmission has used the 24-dimensional “Leech lattice” because it is the densest possible lattice.

An -dimensional ball has almost all its volume very near the surface. The volume of a ball of radius is times the volume of a unit ball. So if , and if , then the -radius ball has less than 35 percent of the volume of the unit-radius ball, so about 65 percent is within 1/10 of the distance to the spherical surface.

Concentration of measure

It is now known there is a much more general and counter-intuitive high-dimensional effect called concentration of measure (CoM) [62], [124].

A real-valued function on a “nice” metric space is Lipschitz if . The phenomenon of CoM is that every Lipschitz function is highly concentrated about its median value.

For example, let be a positive integer and consider the unit sphere of all points in with Euclidean-distance 1 from the origin. Define a function by choosing one of the axes as “vertical” and putting

equal to the fraction of the area of the sphere under the hyperplane which intersects the chosen axis at

. So , , and . The median value is achieved by (any) hemisphere by symmetry. Let denote the -expanded region consisting of all points on the sphere within (angular distance) of . Once dimension is sufficiently large, the area of is almost ! (Reversing the two hemispheres, one concludes that almost all the area of the sphere is within of any equator.) According to Ledoux’s well-written notes [62, p 18], this instance of dimensional concentration was observed by Paul Lèvy in 1919.

This is consistent with a remark by Richard Hamming [46, p 188]. Let be a given small positive number. Call two vectors -almost perpendicular if the angle between them differs from by at most . Then for sufficiently large and a fixed vector in the bipolar (Hamming) cube , there are almost vectors such that and are -almost perpendicular. (Observe that the Euclidean norm of the points in is .) Using more subtle arguments [52], it can be shown that contains a subset of elements which are pairwise -almost perpendicular. If and , this is about 403 elements (instead of 108) and the ratio increases rapidly with . For instance, if , then the resulting “quasiorthogonal” set has approximately 162,755 elements. (We are ignoring, for simplicity, the difference between Euclidean distance and angular distance.) Moreover, Gorban and Tyukin have shown [36] that such quasiorthogonal sets are ubiquitous!

In modern datasets, measured dimension of the data vastly exceeds the cardinality of datasets; datapoints lie on the boundary of their convex hull, opposite to the situation in 2-d [24]. Why does that happen?

Data-dimension is rapidly increasing due to new technology, but number of data-points in a given data-set has stayed nearly constant - especially in connection with medicine and other fields where legal concerns are tracking each element. Another reason for the slow growth of data-set cardinalities is the increasing effort of the many individual measurements. Hence, .

Thus, we find in a natural way zero-one phenomena such as the following: the area of a polar cap of radius exactly (corresponding to a hemisphere) is exactly 1/2 but a cap of radius has area , while a cap of radius has area . This means that the expected area of a cap with radius plus small zero-mean noise is but almostly surely any particular cap will have area 0 or area 1.

Similarly, in large dimension, balls (everything within a certain radius of the origin) have volume almost zero or almost infinity depending on the radius. Such zero-one laws in probability theory, where events are almost certain or almost forbidden, were already known to Kolmogorov

[56, pp 69–70].

In the subsections below, we sketch some of the applications of this phenomenon to the engineering issues of Lunaport.

Concentration of measure applied to machine learning

Like optimal transport theory, concentration of measure may make its main contribution to engineering in related computations. CoM phenomena have been applied to neural networks by Kurkova & Sanguineti [57] and by Gorban & Tyukin [36], while Lévy & Schwindt [66]

have applied transport theory to geometric computation. I believe that these approaches will lead to better pattern recognition and, thus, ultimately to autonomous robots.

Concentration phenomena may also be helpful in finding good solutions to decision and planning problems where the universe of choices is very large as one may expect in a universe of Big Data, where it is utility which is being transported.

Implicit assumptions

An implicit assumption has been that dimension is an integer! Unlike the theory of gears with an irrational number of teeth (in an apocryphal joke), fractional dimensions turn out to be very useful! For instance, they are perfect for generating visual textures (fractals), and we suspect also for designing new metamaterials such as that proposed above for a meteor shield. Fractional dimensons are involved in research on flow [23] and networks [107].

This utility of math should not be surprising; see Eugene Wigner’s essay The unreasonable effectiveness of mathematics in the natural sciences [131].

Math and mechanics

The connection between mathematics and mechanics was the guiding vision of Galileo, Newton, and the Enlightenment (see, e.g., Ekeland [29]). This sort of mathematics isn’t just theoretical - it is demonstrable. For instance, comparing a straight-line ramp and a longer curved ramp, which follows the cycloid curve, where curved and straight ramps start and finish side-by-side, one sees that if two wheeled objects are released simultaneously, the one which travels farther arrives first! In fact, the cycloid beats any other choice.

The math involved can be quite nontrivial - for instance, the calculus of variations. Even simple rotational motion has surprises. Felix Klein (a founder of topology) wrote a treatise on the top, a child’s toy based on the gyroscopic principle, where he employs four complex variables, including complex time, to obtain an efficient representation. The much more mundane Hamilton theory of quaternions is widely used in the control of rotation due to its superior speed of calculation. It also avoids gymbal lock.

A modern example of the harmony of math and mechanics is the so-called spin-back. The author saw one demonstrated at a math meeting in California, where a JPL engineer showed a block of wood which had been whittled into a smooth but asymmetric form. When spun clockwise on a table the object slowed, stopped spinning, commenced to rock back-and-forth, and then finally began to spin counter-clockwise (slowly). We examined the piece of wood, and it had normal weight and felt solid.

Mechanical devices have been used and designed for calculation. Among others, these include the compass, straight-edge, and T-square; the slide-rule and sun-dial; the abacus and Babbage’s difference engine; and last but not least specialized analogue computers, including optical calculation of Fourier and Fresnel transforms using holographic lenses. Indeed, Kolmogorov (one of the inventors of neural networks NN) observed that they also constitute a kind of nomographic device.

Optimal Transport Theory

In mathematics and economics, transport theory is the study of optimal transportation and resource allocation. The problem was formalized by the French mathematician Gaspard Monge in 1781, and was significantly expanded in the 20th century by Kantorovich, Hitchcock, and others. Recently, theory has undergone enhancements (e.g., Villani [127], Cuturi [19]). This sophisticated theory applies only to the least complex transport, dealing with essentially fungible substances, but it allows both discrete and continuous “flavors” as well as a “semi-continuous” computer implementation [66].

However, if all one cares about is mass of, e.g., coal, rather than its detailed qualities, then transport theory is fine. It might be useful to optimize the logistics of mining materials from the lunar surface, processing them, and then placing them appropriately to build a dome or other structure. Transport theory may turn out to be useful for autonomous robots - e.g., in real-time image processing algorithms, as Haker, Zhu, Tannenbaum, & Angenent [45] have shown how optimal transport can be used to put multiple images of the same object into register via adaptive warping.

Algebraic approaches

One approach to understanding biological and the nature of complex systems has been to impose the structure of mathematical category theory, which is at its core a theory of diagrams. Category theory allows explicit enrichment of mathematical objects through the emergence of new and sometimes unexpected properties of what one might call “gratuitous coherence.” For instance, Mac Lane [71] showed that commutativity holds for all diagrams of a certain type if it holds for a particular explicit (small) subset of such diagrams.

The theory of Ehresmann and Vanbremeersch (category theorist and medical scientist) [28] (cf. [51]) fits this framework. I think that having machine intelligence capable of checking the assertions for multidimensional diagrams could permit the EV evolutionary category model to be instantiated by an AI.

Design

Mathematical concepts are attractive for design because they can create optimal systems - indeed, without some precise framework, the notion of optimality is without meaning. Math can also help in the analysis of really large problems as it provides a principled way to understand, describe, and simulate future engineered structures.

While bridges and building construction have shown for centuries that brilliant and energetic engineers and architects can solve the problems posed to them, in recent years the accumulation of regulations and bureaucracy have greatly increased cost without a commensurate improvement in the result. Thus, in building a lunar spaceport and more generally in industrializing the moon, it may be helpful to systematize the approach and to use a combination of formal methods with pragmatic implementation.

10 The transdisciplinary problem of space

For the reward of unlimited energy and material resources and an existential new frontier, we must coordinate a tapestry of rapidly evolving technologies. Ordinary interdisciplinary cooperation runs the risk (indeed, the certainty) of deadly disconnects such as having shafts that spin clockwise trying to drive propellors that spin counter-clockwise. In the harsh reality of space, mistakes can be fatal. Thus, it is important that the collaboration between specialties be of sufficient depth and duration as recommended by Ivan Havel who coined the term “transdisciplinary” in order to draw attention to the issue.

In the current environment, it should be clear to any introspective person that the consensus view isn’t necessarily correct, that emotions can cloud judgement, and that nature can surprise us. All the more reason for an open-minded, comprehensive study of our options in constructing the transport facility on the moon we have been calling Lunaport.

There are a few precursors to our study such as “compact cities” by George Dantzig and Thomas Saaty in the early 1970s [20] and even earlier in science fiction, e.g., the Cities-in-Flight series [9] from the 1950s and early 60s.

Eventually we will be ready for the next step in humanity’s exploration of space - the leap to the stars. At this point, we are still learning about the composition of the Interstellar Medium (ISM) [59], but visionaries like Robert W. Bussard imagined a means to travel vast distances by collecting the ISM and ejecting it out the back of the space craft [12]. Bussard’s original scheme (from 1960) used magnetic fields to scoop in the ISM, compressing it so intensely as to create nuclear fusion from the hydrogen. But perhaps something else could achieve the needed energy, with the ISM playing a passive role. A nuclear chain reaction, producing fission, requires a sufficiently dense flux of neutrons, but a plasma of ionized uranium nuclei might be made to achieve the threshold density in a small region and periodically - a thousand miniature atomic bombs going off each second. This is a cross between “Ram Augmented Interstellar Rocket” described in [12] and nuclear pulse propulsion (as in the Orion project) [114].

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