Polygonal Dissections - Penrose Tilings

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Deposit #89

Greg Frederickson
Purdue University

gnf@cs.purdue.edu

A Penrose Sampler

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Wadham College, Oxford, one of the newest in the 17th century. Recently Sir Roger Penrose retired there.

In a garden in the back of the College, a patio has been tiled with a design based on Sir Roger's work.

Computer animations now enhance centuries old mathematical constructions.

Tartaglia's stellated truncated hexahedron from his 1543 edition of Euclid's "Elements."

Polygonal Geometry
Penrose Tilings ~ Hinged Dissections

This section . . . .

Greg Frederickson illustrates the classic hinged dissection of a triangle turned into a square.
Click on the RealPlayer to see his animation.

More Animations
(The animations are under copyright by Greg Frederickson and may not be copied, electronically or otherwise, without his express written permission.)

On catching a first glimpse of the "Water Cube" at the Beijing Summer Olympics in 2008, students remarked they thought they had seen Penrose Tilings. The National Aquatic Center, located in the Olympic Green, became the focal point of American viewers watching Michael Phelps win his record setting eight gold medals. The design by ARUP was not inspired by Sir Roger Penrose but by the somewhat more relevant geometric structures of soap bubbles. Nevertheless the translucent recyclable panels provided a fascinating visual impact, especially at night.

Almost simultaneously other students read Greg Frederickson's article in The College Mathematics Journal on hinged dissections. His opening sentence was prophetic:

"Some mathematical ideas are just so nifty that people can't resist finding a physical realization for them."

Frederickson discusses the history of several mathematicians who have investigated dissections including the above realPlayer animation.

He has also gone on to apply his carpentry skills to construct models of Penrose Tilings.

Howard Eves and others have built sets of four tables based on the hinged design.

A brief list of references that should be in most university libraries.

From the author of Deposit # 89 . . .
Frederickson, Greg N., Designing a Table Both Swinging and Stable, The College Mathematics Journal, 39 (4) September 2008, pp. 258-266.

Frederickson, Greg N., Dissections: Plane and Fancy, Cambridge University Press, 1997.

Weisstein, E. W., CRC Concise Encyclopedia of Mathematics, CRC Press, 1999, pp. 471 - 473. See Dissection.

and from The College Mathematics Journal, 40(1) January, 2009, p. 71,

Julie Rehmeyer on the mathematics of the "Water Cube" in Beijing, 2008 Olympics:
< http://www.sciencenews.org/view/generic/id/34283/description/A_building_of_bubbles >

The National Curve Bank thanks Greg Frederickson for Deposit #89.
Be sure to see his article in The College Mathematics Journal for more information.

< gnf@cs.purdue.edu >