New Properties and Invariants of Harmonic Polygons

Via simulation, we discover and prove curious new Euclidean properties and invariants of the Poncelet family of harmonic polygons.

A Web of Confocal Parabolas in a Grid of Hexagons

If one erects regular hexagons upon the sides of a triangle $T$, several surprising properties emerge, including: (i) the triangles which flank said hexagons have an isodynamic point common with $T$, (ii) the construction can be extended iteratively, forming an infinite grid of regular hexagons and flank triangles, (iii) a web of confocal parabolas with only three distinct foci interweaves the vertices of hexagons in the grid. Finally, (iv) said foci are the vertices of an equilateral triangle.

Discovering new Properties and Invariants of Harmonic Polygons

Via simulation, we discover and prove curious new Euclidean properties and invariants of the Poncelet family of harmonic polygons.

Hexagonal Flanks, Confocal Parabolas, and a Focal Equilateral

We describe curious properties of a tiling of "flank" triangles and regular hexagons including (i) a common isodynamic point over all flanks, (ii) conservations among sextets of flanks around a fixed regular hexagon, (iii) the ability to build an infinite grid or tiling, and (iv) families of confocal parabolas woven into the tiling, whose (v) three distinct foci are the vertices of an equilateral.

Properties of Parabola-Inscribed Poncelet Polygons

We investigate properties of Poncelet $N$-gon families inscribed in a parabola and circumscribing a focus-centered circle. These can be regarded as the polar images of a bicentric family with respect to the circumcircle, such that the bicentric incircle contains the circumcenter. We derive closure conditions for several $N$ and describe curious Euclidean properties such as straight line, circular, and point, loci, as well as a (perhaps new) conserved quantity.

Loci of Poncelet Triangles with Multiple Caustics

We analyze loci of triangles centers over variants of two-well known triangle porisms: the bicentric family and the confocal family. Specifically, we evoke a more general version of Poncelet's closure theorem whereby individual sides can be made tangent to separate caustics. We show that despite a more complicated dynamic geometry, the locus of certain triangle centers and associated points remain conics and/or circles.

A Theory for Locus Ellipticity of Poncelet 3-Periodic Centers

We propose a theory which predicts the ellipticity of a triangle center's locus over a Poncelet 3-periodic family. We show that if the triangle center can be expressed as a fixed affine combination of barycenter, circumcenter, and a third, stationary point over some family, then its locus will be an ellipse. Taking billiard 3-periodics as an example, the third point is the mittenpunkt. We derive conditions under which a locus degenerates to a segment or is a circle. We show a locus turning number is either plus or minus 3 and predict its movement monotonicity with respect to vertices of the 3-periodic family. Finally, we derive a (long) expression for the loci of the incenter and excenters over a generic Poncelet 3-periodic family, showing they are roots of a quartic. We conjecture (i) those loci are convex, and (ii) that they can only be ellipses if the pair is confocal, i.e., within a 1d subspace of the 5d space of ellipse pairs which admit 3-periodics.

A Theory for Locus Ellipticity of Poncelet 3-Periodics

We provide a theory as to why the locus of a triangle center over Poncelet 3-periodics in an ellipse pair is an ellipse or not. For the confocal pair (elliptic billiard), we show that if the center can be expressed as a fixed affine combination of barycenter, circumcenter, and mittenpunkt (which is stationary over the confocal family), then its locus will be an ellipse. We also provide conditions under which a particular locus will be a circle or a segment. We also analyze locus turning number and monotonicity with respect to vertices of the 3-periodic family. Finally we write out expressions for the convex quartic locus of the incenter for a generic Poncelet family, conjecturing it can only be an ellipse if the pair is confocal.

Poncelet Plectra: Harmonious Properties of Cosine Space

Poncelet N-periodics in a confocal pair (elliptic billiard) conserve the same sum of cosines as their affine image with fixed incircle. For N=3, the vector of cosines in either family sweep the same planar curve: an equilateral cubic resembling a plectrum (guitar pick). We also show that the family of excentral triangles to the confocal family conserves the same product of cosines as its affine image with fixed circumcircle. In cosine space cosine triples in either family sweep the same spherical curve. The associated planar curve in log cosine space is also plectrum-shaped, though rounder than the one swept by its parent confocal family.