There are currently 422 dissections at the Wolfram Demonstrations site, so I thought I’d give a talk about them. Here are the records as they stand at Gavin Theobald’s Polygon Dissections.
I was also recently intrigued by Quanta’s Animation for a six fractal piece square to circle dissection by Andras Máthé, Oleg Pikhurko and Jonathan Noel.
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"]
Triangle to Pentagon
ResourceFunction[“CreditedDemonstration”][“GoldbergsPentagonToTriangleDissection”]
We can also get close, as in Pentagon-to-Near-Equilateral-Triangle Dissection.
Triangle to Hexagon
ResourceFunction[“CreditedDemonstration”][“LindgrensHexagonToTriangleDissection”]
Triangle to Heptagon
Theobald’s Dissection of a Heptagon to a Triangle
Triangle to Octagon
Theobald’s Dissection of an Octagon into a Triangle
Triangle to Nonagon
Theobald’s Nonagon-to-Triangle Dissection and Hanegraaf’s Dissection of a Nonagon into a Triangle.
All of these can be laser cut or 3D printed, as shown by Izidor Hafner.
Square to Pentagon
Freese’s Dissection of a Regular Pentagon to a Square
ResourceFunction["CreditedDemonstration"]["BrodiesPentagonToSquareDissection"]
ResourceFunction["CreditedDemonstration"]["PentagonToNearSquareDissection"]
Adam Gsellman: Pentagon to rectangle is 4.
Square to Hexagon
Busschop’s Hexagon-to-Square Dissection
ResourceFunction["CreditedDemonstration"]["BradleysHexagramToSquareDissection"]
Square to Heptagon
Theobald’s Heptagon-to-Square Dissection
Square to Octagon
Bennett’s Octagon-to-Square Dissection
Square to Nonagon
Theobald’s Nonagon-to-Square Dissection
Square to Decagon
Decagon-to-Square Dissection
Square to Dodecagon
Freese’s Dissection of a Regular Dodecagon into a Square
Lindgren’s Square-to-Dodecagon Dissection
Pentagon to Hexagon
Lindgren’s Dissection of a Pentagon to a Hexagon
Pentagon to Heptagon
Theobald’s Dissection of a Heptagon to a Pentagon
Hexagon to Heptagon
Theobald’s Heptagon-to-Hexagon Dissection
Hexagon to Dodecagon
Dodecagon-to-Hexagon Dissection
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"]
Regular polygons with an edge length of 2 have some odd properties.
- What is the area of the annulus?
- What is the square distance sum between vertex heights?
The results of Shattering the Plane came out of a simple question: how can similar triangles surround a point? So far, this question hasn’t been answered in 3D with tetrahedra.
That particular dissection leads to the following substitution tiling:
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 0.
There are 3D dissections.
Hanegraaf’s Rectangular Block of Size 211 to a Cube Dissection
Most dissections lack minimality proofs!
hexy = CirclePoints[6];
Graphics[{Line[Join[hexy, {hexy[[1]], hexy[[3]], Mean[{hexy[[1]], hexy[[3]]}], hexy[[2]]}]]}]
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?