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A web of confocal parabolas in a grid of hexagons

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Abstract

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.

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Notes

  1. This pencil is known as the Schoutte pencil (Johnson 1917).

  2. This is the double-length reflection about the barycenter \(X_2\).

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Acknowledgements

We thank Arseniy Akopyan for contributing crucial insights and discovering the web of parabolas in the grid. Darij Grinberg lend us an attentive ear and proved the stationarity of \(X_{16}\). Clark Kimberling was kind enough to include our early observations on the Encyclopedia of Triangle Centers.

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Authors and Affiliations

  1. Moparmatic Inc., Worcestershire, England

    Peter Moses

  2. Data Science Consulting Ltd., Rio de Janeiro, Brazil

    Dan Reznik

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  1. Peter Moses
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  2. Dan Reznik
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Correspondence to Dan Reznik.

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Appendix A: Explicit formulas

Appendix A: Explicit formulas

To facilitate computational reproduction of our results, we provide code-friendly expressions for a few objects. In the expressions below a, b, c refer to sidelengths and x, y, z are barycentric coordinates.

1.1 A-parabola

It is given implicitly by:

figure a

The B- and C-parabolas can be obtained by cyclic permutations, i.e., \((a,b,c)\rightarrow (b,c,a)\), and \((a,b,c)\rightarrow (c,a,b)\), respectively.

1.2 The two “skip-1” parabolas

The skip-1 parabola through B is given by:

figure b

The skip-1 parabola through C is obtained with a bicentric substitution, i.e., \((a,b,c,x,y,z)\rightarrow (a,c,b,x,z,y)\).

1.3 Center (at infinity) of the A-parabola group

The A-parabola and BC skip-1 pair of parabolas have parallel axes through \(f_a,f_b,f_c\), therefore their axes will cross the line at infinity at the same point given by the following barycentrics;

figure c

1.4 Directrix of the A-parabola

Is is the line given by:

figure d

The other two directrices can be obtained via cyclic substitution.

1.5 Directrix equilateral

The barycentrics x, y, z of the A-vertex of the directrix equilateral (Proposition 12) are given by:

figure e

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Moses, P., Reznik, D. A web of confocal parabolas in a grid of hexagons. Beitr Algebra Geom 64, 669–688 (2023). https://doi.org/10.1007/s13366-022-00651-1

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