## 1 Introduction

The front doors of apartment buildings lining many a street in Rio de Janeiro, Brazil, are bedecked with wrought iron^{1}^{1}1Iron was popular for ornamental applications in the 19th and early 20th century, given its high ductility and structural capabilities. It has now been by and large replaced by steel. masterpieces. Their curvaceous designs seem to strike a perfect balance between beauty and sturdiness.

As one experiments with the curves swept by points attached to Poncelet triangle families (more details in Section 2), one becomes aware of certain common motifs linking these two worlds, e.g., four-fold symmetry, harmonious spirals, tiling, recursion, etc., see Figure 1.

We had built a web-based tool to interact with Poncelet triangles and their loci, see [5, 30]. As new experiments were set up, we would often stumble upon a new ornate locus. We quickly recognized in Poncelet loci a kind of aesthetic talent. Gradually, we added features geared at beautifying, coloring, and sharing such curves, see Figure 2. At the same time, we started to collect hundreds of aesthetically-pleasing finds into slideshow and video “galleries” [27, 28, 29]. Leafing through these gives one an idea of the wide design palette possible with Poncelet loci.

Our goal here is to explore the ingredients involved in generating such loci. In the next sections we (i) review Poncelet’s porism, (ii) review the basic geometry of a triangle’s notable points (whose loci we will sweep), (iii) explore some features of our locus-rendering app, and (iv) tour a few the artful byproducts of Poncelet loci.

### Related work

The fields of kinetic art [23] and computer-generated (generative) art [17] have been evolving for several decades. Works that explore the connection between classical art and geometry include [14, 18, 20]. The usage of dynamic geometry tools for mathematical discovery is beautifully expounded in [33]. In [21, 22, 34, 37, 38] loci of triangle centers are studied over various 1d triangle families, Poncelet or otherwise. Works [25, 24] follow in their footsteps and identify new curious properties and invariants of Poncelet families. Proofs that loci of certain triangle centers in over the confocal family are ellipses appear in [8, 12, 9, 32]. A theory of locus ellipticity is slowly emerging, see [15, 16]. In [10, 11, 26], similar geometric properties and invariants are used to cluster Poncelet triangles families. Proofs of experimentally-detected invariants of Poncelet families appear in [1, 2, 4, 35].

## 2 Loci of Poncelet Triangles

Poncelet’s closure theorem is illustrated in Figure 3. It states that given two real conics^{2}^{2}2Recall these comprise ellipses, hyperbolas, parabolas, as well as some degenerate specimens [13]. , if one can draw a polygon with all vertices on and with all sides tangent to , then a one-dimensional family of such polygons exists. [6, 7, 3].

We will herein focus on families of Poncelet triangles families. In Figure 4, six examples are shown of such families interscribed^{3}^{3}3This is shorthand for “inscribed while simultaneously circumscribing”. between two concentric, axis-aligned ellipses^{4}^{4}4In general, the pair of Poncelet conics need not be concentric nor axis aligned..

Referring to Figure 5, a first natural question is: over some particular triangle family, what are curves swept by a notable point? Recall the four classical notable points of a triangle, shown in Figure 6, namely, (i) the incenter , (ii) the barycenter , (iii) the circumcenter , and (iv) the orthocenter . The notation conforms with [19], where thousands of such points, known as triangle centers, are specified.

Referring to Figure 6(left), an early observation was that over the confocal family, the loci of the four notable points are ellipses. Formal proofs appeared in [32, 8, 12]. Interestingly, other centers can sweep quartics, self-intersecting curves, segments, and be stationary points, see Figure 6(right), and [9].

## 3 The Locus Rendering App

Referring to Figure 8, the default view of our tool is the elliptic locus of the incenter over the confocal family. The app can render loci of the first 1000 triangle centers in [19], over one dozen Poncelet families, including the ones of Figure 3. The user can interactively change parameters of the simulation (e.g., aspect ratio of Poncelet ellipses, triangle center tracked, derived triangle being used, etc.), and observe topological changes in the loci being studied. As an example see Figure 9.

Loci of triangle centers of derived triangles can be studied as well, e.g., the orthic, medial excentral, triangles^{5}^{5}5These correspond to triangles with vertices which are the feet of altitudes, medians, and where external bisectors meet, respectively., etc., see [36, 19]. As shown in Figure 10, the locus of vertices of the main or derived triangles can be studied as well, further expanding the palette of obtainable curves.

## 4 Aestheticizing Poncelet

As a mathematical discipline travels far from its empirical source, or still more, if it is a second and third generation only indirectly inspired from ideas coming from reality, it is beset with very grave dangers. It becomes more and more purely aestheticizing, more and more purely l’art pour l’art. – John von Neumann, “the Mathematician”

Not exactly following von Neumann’s advice, we added features to our locus tool to beautify loci: dark backgrounds, thick outlines, and automatic color-filling of connected regions with a random palette of pastel colors, see Figures 12, 11 and 13. As mentioned above, 100s of such designs are now collected in [27, 28].

### Interaction with a digital designer

## 5 Conclusion

Art, architecture, and design have been inspired by geometry and mathematics and vice-versa. Common design motifs between wrought iron gates and Poncelet loci have stimulated us to look at the latter both from a geometric and an aesthetic perspective. As seen in Figure 16, wrought iron kraftwerk is on a league of its own. An interesting question is if Poncelet loci could ever be used as a basis for new metalwork and/or architectural design.

### Acknowledgements

I am grateful to my sister Regina Reznik for her artistic design work, and Iverton Darlan for being a co-developer of the locus app.

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