# Polygon Dissections **URL:** \ **Category:** Ideas\ **Tags:** demonstrations, geometry, graphics-and-visualization, interface-construction, events-and-media\ **Created:** [August 17, 2023, 6:03pm UTC](https://community.wolfram.com/t/polygon-dissections/22667 "2023-08-17T18:03:46Z")\ **Posts on this page:** 1\ **Page:** 1
**Author:** ![edp](https://yyz1.discourse-cdn.com/wolfram/user_avatar/community.wolfram.com/edp/32/292_2.png) [@edp](https://community.wolfram.com/u/edp)\ **Post date:** [August 17, 2023, 6:03pm UTC](https://community.wolfram.com/t/polygon-dissections/22667/1 "2023-08-17T18:03:46Z")
![temp.jpg](https://global.discourse-cdn.com/wolfram/original/3X/f/b/fb0024fba3df7dc6940825dc11a272fc169a3010.jpeg) There are currently 422 [dissections](https://demonstrations.wolfram.com/search.html?query=dissection "dissectionsquot;") at the [Wolfram Demonstrations](https://demonstrations.wolfram.com/) site, so I thought I’d give a talk about them. Here are the records as they stand at [Gavin Theobald’s Polygon Dissections](http://www.gavin-theobald.uk/HTML/Index.html). ![temp2.jpg](https://global.discourse-cdn.com/wolfram/original/2X/8/86b8f5cdbb5538ed5ea201857cf8db3fc9dbd5c9.jpeg) I was also recently intrigued by [Quanta’s Animation](https://www.quantamagazine.org/an-ancient-geometry-problem-falls-to-new-mathematical-techniques-20220208/) for a six fractal piece square to circle dissection by Andras Máthé, Oleg Pikhurko and Jonathan Noel. ![temp3.jpg](https://global.discourse-cdn.com/wolfram/original/3X/c/0/c0863a50773531ad6b676d02c887895a8f0ced4c.jpeg) The following code shows two polygons with the same unit area. ``` UnitAreaPolyPoints[k_, angle_ : 0] := CirclePoints[{Sqrt[2] Sqrt[Csc[(2 \[Pi])/k]/k], angle }, k]; Manipulate[Graphics[{ Line[Append[#, First[#]] &@UnitAreaPolyPoints[j, anglej]], Line[Append[#, First[#]] &@UnitAreaPolyPoints[k, anglek]]}], {{j, 3}, 3, 12, 1}, {{k, 5}, 3, 12, 1}, {{anglej, 0}, -Pi, Pi}, {{anglek, 0}, -Pi, Pi}, SaveDefinitions -> True, SynchronousInitialization -> False] ``` With this, it’s possible to prove that a triangle to square dissection requires at least 3 pieces. New we can start showing dissections. **Triangle to Square.** ``` ResourceFunction["CreditedDemonstration"]["EquilateralTriangleToSquare"] ResourceFunction["CreditedDemonstration"]["HaberdashersProblem"] ResourceFunction["CreditedDemonstration"]["AnExampleOfTheBolyaiGerwienTheorem"] ResourceFunction["CreditedDemonstration"]["DissectionOfASphericalTriangleIntoASquare"] ``` ![temp4.jpg](https://global.discourse-cdn.com/wolfram/original/3X/2/6/26918f7088be664be2434dbcafac6d7d03f934bc.jpeg) **Triangle to Pentagon** ResourceFunction[“CreditedDemonstration”][“GoldbergsPentagonToTriangleDissection”] We can also get close, as in [Pentagon-to-Near-Equilateral-Triangle Dissection](https://demonstrations.wolfram.com/PentagonToNearEquilateralTriangleDissection/). ![temp5.jpg](https://global.discourse-cdn.com/wolfram/original/3X/2/6/26ab159a8d5a9ecf521ab4c437d792c8bce67aab.jpeg) **Triangle to Hexagon** ResourceFunction[“CreditedDemonstration”][“LindgrensHexagonToTriangleDissection”] **Triangle to Heptagon** [Theobald’s Dissection of a Heptagon to a Triangle](https://demonstrations.wolfram.com/TheobaldsDissectionOfAHeptagonToATriangle/) **Triangle to Octagon** [Theobald’s Dissection of an Octagon into a Triangle](https://demonstrations.wolfram.com/TheobaldsDissectionOfAnOctagonIntoATriangle/) ![temp6.jpg](https://global.discourse-cdn.com/wolfram/original/2X/c/c08c1007d74b2677c2ef48ed53b2d810dd4345ef.jpeg) **Triangle to Nonagon** [Theobald’s Nonagon-to-Triangle Dissection](https://demonstrations.wolfram.com/TheobaldsNonagonToTriangleDissection/) and [Hanegraaf’s Dissection of a Nonagon into a Triangle](https://demonstrations.wolfram.com/HanegraafsDissectionOfANonagonIntoATriangle/). ![temp7.jpg](https://global.discourse-cdn.com/wolfram/original/2X/7/77a6f5a0bea71bceb593f7128032c84aae7c3b5a.jpeg) All of these can be laser cut or 3D printed, as shown by Izidor Hafner. ![dissect.jpg](https://global.discourse-cdn.com/wolfram/original/2X/4/45612354308584f227b2ccaa298a1905c8d22d3e.jpeg) **Square to Pentagon** [Freese’s Dissection of a Regular Pentagon to a Square](https://demonstrations.wolfram.com/FreesesDissectionOfARegularPentagonToASquare/) ![temp8.jpg](https://global.discourse-cdn.com/wolfram/original/3X/3/a/3afcb32b14ce6eb2d5b2b55cc8a69d046e5e8dae.jpeg) ``` ResourceFunction["CreditedDemonstration"]["BrodiesPentagonToSquareDissection"] ResourceFunction["CreditedDemonstration"]["PentagonToNearSquareDissection"] ``` Adam Gsellman: Pentagon to rectangle is 4. ![temp9.jpg](https://global.discourse-cdn.com/wolfram/original/2X/f/f1b82fe76ec07a5e5793272a02ec898b2ed406ab.jpeg) **Square to Hexagon** [Busschop’s Hexagon-to-Square Dissection](https://demonstrations.wolfram.com/BusschopsHexagonToSquareDissection/) ``` ResourceFunction["CreditedDemonstration"]["BradleysHexagramToSquareDissection"] ``` ![temp10.jpg](https://global.discourse-cdn.com/wolfram/original/3X/9/2/92c2c3d867d51a5e6747b37f6614e3f9b436c9c7.jpeg) **Square to Heptagon** [Theobald’s Heptagon-to-Square Dissection](https://demonstrations.wolfram.com/TheobaldsHeptagonToSquareDissection/) ![temp11.jpg](https://global.discourse-cdn.com/wolfram/original/2X/4/4c4fc809af9553dcb58d33c2bb2d468e92c79eee.jpeg) **Square to Octagon** [Bennett’s Octagon-to-Square Dissection](https://demonstrations.wolfram.com/BennettsOctagonToSquareDissection/) ![temp12.jpg](https://global.discourse-cdn.com/wolfram/original/1X/eddd1c9f78d362c7f6bf609abec95c8422ecf15f.jpeg) **Square to Nonagon** Theobald’s Nonagon-to-Square Dissection ![temp13.jpg](https://global.discourse-cdn.com/wolfram/original/3X/8/e/8ec32dc9e6ce321bb86bc0fd1d9fc1cb37660ef0.jpeg) **Square to Decagon** [Decagon-to-Square Dissection](https://demonstrations.wolfram.com/DecagonToSquareDissection/) ![temp14.jpg](https://global.discourse-cdn.com/wolfram/original/3X/4/c/4c17ffdda78eb19882a54a9485fa221cc7753b00.jpeg) **Square to Dodecagon** [Freese’s Dissection of a Regular Dodecagon into a Square](https://demonstrations.wolfram.com/FreesesDissectionOfARegularDodecagonIntoASquare/) [Lindgren’s Square-to-Dodecagon Dissection](https://demonstrations.wolfram.com/LindgrensSquareToDodecagonDissection/) ![temp15.jpg](https://global.discourse-cdn.com/wolfram/original/3X/6/2/6282c4c0c2c9c501493cb820c8cba738cda95c8f.jpeg) **Pentagon to Hexagon** [Lindgren’s Dissection of a Pentagon to a Hexagon](https://demonstrations.wolfram.com/LindgrensDissectionOfAPentagonToAHexagon/) **Pentagon to Heptagon** [Theobald’s Dissection of a Heptagon to a Pentagon](https://demonstrations.wolfram.com/TheobaldsDissectionOfAHeptagonToAPentagon/) **Hexagon to Heptagon** [Theobald’s Heptagon-to-Hexagon Dissection](https://demonstrations.wolfram.com/TheobaldsHeptagonToHexagonDissection/) ![temp17.jpg](https://global.discourse-cdn.com/wolfram/original/2X/3/355fbcdb3717b786b61f80bdf808170953685184.jpeg) **Hexagon to Dodecagon** [Dodecagon-to-Hexagon Dissection](https://demonstrations.wolfram.com/DodecagonToHexagonDissection/) ![temp16.jpg](https://global.discourse-cdn.com/wolfram/original/2X/3/32a52dce5f7485689b990bf0510feb7cfd741b62.jpeg) That shows quite a few ways of transforming one polygon into another. It’s also possible to take multiple copies of a polygon and make a larger version of the same polygon. ``` ResourceFunction["CreditedDemonstration"]["DissectionOfOneRegularHeptagonIntoSeven"] ResourceFunction["CreditedDemonstration"]["DissectionOfARegularHeptagonIntoEight"] ResourceFunction["CreditedDemonstration"]["DissectionOfARegularDodecagonIntoThree"] ResourceFunction["CreditedDemonstration"]["DissectionOfARegularDodecagonIntoSixSmallerOnes"] ``` ![temp18.jpg](https://global.discourse-cdn.com/wolfram/original/3X/1/f/1fa87f20ca72a694f554ea4606ef0ff8292048c8.jpeg) Regular polygons with an edge length of 2 have some odd properties. 1. What is the [area of the annulus](https://demonstrations.wolfram.com/AnnulusForARegularPolygon/)? 2. What is the [square distance sum](https://demonstrations.wolfram.com/RegularPolygonsOfEdgeLengthTwo/) between vertex heights? The results of [Shattering the Plane](https://blog.wolfram.com/2019/03/07/shattering-the-plane-with-twelve-new-substitution-tilings-using-2-phi-psi-chi-rho/) came out of a simple question: how can [similar triangles surround a point](https://demonstrations.wolfram.com/WheelsOfPoweredTriangles/)? So far, this question hasn’t been answered in 3D with tetrahedra. ![temp19.jpg](https://global.discourse-cdn.com/wolfram/original/2X/c/cb0bed1e1b6b4206e39a7d26c4c80f321a4af3d5.jpeg) That particular dissection leads to the following substitution tiling: ![temp20.jpg](https://global.discourse-cdn.com/wolfram/original/3X/9/d/9d97436e378f208f3be072cf6173fa69379d5b38.jpeg) Wallace[Dash]Bolyai[Dash]Gerwien theorem: Two polygons of the same area can be dissected into each other. Discovered independently by Wallace: 1807, Bolyai: 1833, Gerwein: 1835. Hilbert’s 3rd problem asked about 3D. Max Dehn solved the problem: No. He proved you can’t cut a regular tetrahedron into pieces to make a cube. Polyhedra of the same volume are Scissors Congruent iff they have the same Dehn Invariant. It turns out that the [Engel-38 polyhedron has Dehn invariance](https://wolfram.discourse.group/t/20719/1) 0. There are 3D dissections. [Hanegraaf’s Rectangular Block of Size 2_1_1 to a Cube Dissection](https://demonstrations.wolfram.com/HanegraafsRectangularBlockOfSize211ToACubeDissection/) ![temp21.jpg](https://global.discourse-cdn.com/wolfram/original/3X/c/3/c327d472bb437918e3ed7af47627ba7c4bcc2142.jpeg) Most dissections lack minimality proofs! ``` hexy = CirclePoints[6]; Graphics[{Line[Join[hexy, {hexy[[1]], hexy[[3]], Mean[{hexy[[1]], hexy[[3]]}], hexy[[2]]}]]}] ``` ![temp22.jpg](https://global.discourse-cdn.com/wolfram/original/3X/8/0/802afc8b7c07e061d608eb0b6dd5e54aac3c30e5.jpeg) For example, above, a hexagon is divided into 3 pieces to make a rectangle. Is there a proof that 2 pieces isn’t enough for cutting a hexagon into a rectangle?