In this Section,
22 sets / 2 pages:

Investigating the strange phenomena of five-fold symmetry.
A bit about
pentagon tilings

See gameboard and brief description

... discovered by Sir Roger Penrose, recipient of
Nobel Prize in Physics 2020

... with a great strategy game
by Nancy Van Schooenderwoert

Penrose's quadrilaterals form non-periodic symmetries (no regularly repeating patterns) out to infinity. They're sold by the pod (see at right) of 5 kites, 3 darts, approximating the golden ratio, in assorted colors. All-acrylic tiles measure 1.25 inches on their longer sides. The master set contains 40 pods (320 pieces total, 8 pods each of five colors) plus a handsome 16" vinyl game mat with the grid structure shown at left. For ages 10 to adult, 1 to 5 players.

20-pod starter set, $85
(tiles only, no game)
Add to your order
40-pod master set, $149
(with game supplement)
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8-piece pods, each, $6
Add to your order


If these thick and thin diamonds had special markings on them to force certain adjacencies, they would represent Penrose's patented concept. We've left them blank so they can do Penrose's non-periodic tilings plus attractive kaleidoscopic variations. With 1.25-inch long sides, they're size-compatible with Kites and Darts above and with the Collidescape triangles below. Starter set has 4 pods each in five colors (160 pieces total). Development set has 10 pods each in five colors (400 pieces total). Each pod has 5 thick, 3 thin diamonds (shown at right). These are the tiles seen on the cover of Discover magazine some years ago.

20-pod starter set, $75
Add to your order
50-pod development set, $175
Add to your order
8-piece pods, each, $6
Add to your order

Penrose Kites and Darts and Penrose Diamonds
are made by Kadon under license.

....the golden triangles

Two isosceles triangles have the interesting properties of forming ever larger models of themselves, and of modeling any parts of pentagon tilings. For example, two of the "wide" triangles joined on their long side form a thick Penrose diamond. Joined on their shorter side, they form a "dart." Join two of the taller triangles at their base and you get a thin Penrose diamond. Join them on their longer side and get a "kite". Much of this research was done by Ward Hollins, who named them "Collidescape" for their kaleidoscopic symmetries and non-periodic tiling elusiveness. Each pod: 5 wide, 3 tall triangles with 1.25-inch long sides, equal mix of five colors.

25-pod starter set, $85
Add to your order
55-pod expansion set, $175
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ONESTONETM (Spectre)...
discovered by David Smith, developed with Craig Kaplan,
Joseph Myers, and Chaim Goodman-Strauss

Here is a historical first, sought for centuries and finally found, that has all the math world in an uproar: one shape of tile ("one stone") that can fill the infinite plane with no regularly repeating pattern (non-periodically, below left) without turning over and without holes. Two differently shaped tiles were the known minimum until 2023, when this tile, nicknamed "Spectre", was discovered by David Smith, assisted by three mathematical sages. Why this pursuit? Because all previously known single tiles, like squares or triangles, easily make regular tilings, like checkerboards and wallpaper patterns, or won't fit infinitely. Interesting side note: "onestone" in German is "Einstein". We include links to information collected so far on this phenomenon. The tiles are lasercut acrylic and about 3 inches tall. You can order them by quantity. A handy number is 16 each of 3 colors, black, white, gray (48 pieces), or more for extensive research. Change "1" on the order form to the number desired. If you'd like a fourth color, choose red or blue and indicate how many. Four colors allow "map coloring" with no two of the same color touching. Price per piece,     $1.00

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POCKET STARTM...styled by Kate Jones

This adorable little set of 25 tiles (15 kites, 10 darts) in 5 colors is one of several "25"-based sets created for Kadon's 25th anniversary in 2005. Lets you make hundreds of symmetry patterns with color and shape. Travels easily in its drawstring pouch, so do take it along in pocket or bag. It's a sweet pastime, intriguing icebreaker, creative entertainer, relaxing stress remover, fun to doodle with anytime, anywhere, alone or with a partner. Leaflet suggests several avenues of exploration. For ages 8 to adult. (Shipping discount) $20

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Larger images of Arc Angles

Win a prize!
We're looking for the smallest possible area enclosed by a 25-Arc loop with no overlapping pieces. Send us your best solutions, by email or snailmail.

Originally designed for Kadon's 25th anniversary in 2005, the earliest sets had silver paths on black tiles. Based on the dissection of a pentagon, each arc is 72 degrees. The paths connect any of five points on one edge to any of five points on the other edge, an elegant 25-fold permutation. Five arcs can form a ring, and the whole set can make five separate 4" rings, with neatly matched paths. Or combine 15 arcs into a larger closed loop, or the full 25 into the largest closed loop (see also contest question at left). Lasercut acrylic, handpainted paths, with four little screens to hide your tiles when playing the game, all in drawstring pouch. Sets may have other colors. A curvy challenge for 1 to 4 players, ages 8 to adult. $35

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Pentarose is surely one of our most gorgeous art puzzles. Transformed pentagons and sections of stars are the very shapes that fit together without leaving holes. They are based on prototiles discovered by Sir Roger Penrose during research into non-periodic tilings. We've styled it in four luminous acrylic colors in an 11" circular tray, with a small easel included. Recommended for ages 12 to adult. $72

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Pentarose was developed and is made by
Kadon under exclusive license.

©1998-2024 Kadon Enterprises, Inc.