More about polyforms

Types of sets

Polyforms are phenomenal in how they can fit together to build larger figures, to fill the plane without holes, and to form symmetrical and congruent shapes.

The exploration of polyform puzzles has a long history, dating back to the 1920s. The mathematical names for the various types have usually been introduced by their respective investigators.

Solomon Golomb started the trend in 1953 with polyominoes (see also More about polyominoes and polycubes). These are so rich a subject that we gave them their own section, Polyominoes and Polycubes, in this website.

Shapes made of equilateral triangles became polyiamonds, from diamond, made of two triangles. Shapes made of hexagons are simply polyhexes, and shapes made of isosceles right triangles are polytans, from their resemblance to tangrams. In the literature polytans are sometimes also referred to as "polyaboloes."

Kadon has been making polytan sets since 1988, privately distributed but not offered to the general public, because these sets were thought to be much too difficult for casual fun and recreation. For the determined solver, we are happy to make them available. Here, then, are our Tan Tricks sets, sizes 1 through 6. Their extensive book of activities was published in 2009. If you have the sets but no book, write or email us for a complimentary copy.

A nicely simplified version of this group is the Shape by Shape set we sell in the Puzzles section, made by ThinkFun, Inc. (formerly Binary Arts) from a design by Nob Yoshigahara. Nob used our Tan Tricks during his research.

Dan Klarskov in Denmark is a polyfom enthusiast, blues musician, engineer and woodworker. He is making several polyform sets—polyominos, polyiamonds and polyhexes— in fine wood, shaping the pieces into whimsical animals and offering them as "step" games in progressively longer figures to solve. See them on his website, Klarskov Puzzles.

A fine game company, Brainwright, a division of CEACO, has licensed three of Kadon's puzzles to produce in their own plexiglass versions, sold in game and bookstores such as Barnes & Noble and on Amazon. These three in May 2016 won the prestigious Major Fun Award. Check their site for retailers that carry them: Roundominoes, Iamond Hex and Hexnut Jr. They look like this:


In 2019, Kate finally wrote an article published in Telicom, the quarterly journal of the International Society of Philosophical Enquiry: "A Periodic Table of Polyform Puzzles". It encompassed all the major named categories, illustrated by many of Kadon's sets.

In 2020, for the Gathering4Gardner Celebration of Mind, Kate converted the article into a longer PowerPoint presentation, complete with musical background and rhymed narration. Due to pandemic postponements, it was presented virtually on March 21, 2021—coincidentally World Poetry Day. You can also read it in compact document form, without audio: A Periodic Table of Polyform Puzzles.

Extending the research

Col. George Sicherman's website presents a section on "Polyform Curiosities", investigations into the esoteric problems of exclusion, compatibility and oddities, with examples and solutions.

Nick Maeder, a puzzle genius from New Zealand, has been generating spectacular solutions with large sets. Here is a growing gallery of his hexahex creations.

Solution programs

Solutions for puzzles of this type can now be derived by computer. Aad van de Wetering from the Netherlands has written a fine and quick program for solving figures with polyominoes, polyiamonds and polyhexes. You can download it from his website.

Eric Postpischil has written a nice TriSolve program for solving any polyiamond figure, downloadable from his website.

Andrew Clarke is one of the great polyformists of this century, a legend in his time for amazing, beautiful, difficult and huge solutions. He is now presenting much of this material on a website he calls simply The Poly Page. It's a great reference site, done with style and intelligence.

David Goodger has an enthusiastic polyform project and solving programs on Polyform Puzzler and shows solutions with very attractive colorful graphics. He has generously made his solving program available as freeware and is constantly upgrading it to make it more versatile and adaptable to new types of polyforms as they become defined. Here's a remarkable collection of polyform solutions on the theme of "10" or "X" (Roman numeral for 10) that David designed for the 10th Gathering for Gardner in 2012. See the sample polyiamond figure at right.


And the process of discovery continues. Alan Schoen invented rombiks (Rombix), shapes made of one or two rhombuses joined in circle tilings.

Kate Jones introduced polyrhombs, for polyominoes formed of rhombs rather than squares (Rhombiominoes), and roundominoes (Roundominoes), for circles joined on a square grid. In 2009 Kate introduced polybends, combinations of 1, 2, 3 and 4 quarter arcs (ChooChooLoops).

Henri Picciotto created polyarcs (Polyarcs) formed of curves with a radius the length of the side of a square.

Jacques Ferroul contributed polyspidrons (Poly-Spidrons) that combine triangles and spirals based on Daniel Erdely's trademarked Spidrons. These are by way of being "poly-multiforms," combining more than one kind of building block. Jacques' Stelo is a neat small subset of the highly complex polyspidrons, without the curves.

Jacques also has discovered a huge collection of beautiful shapes to solve with his Tetrapentos, the 7-piece little brother to our Mini-Iamond Ring, using just the tetriamonds and pentiamonds. His website shows many silhouettes as new challenges. If you already own Mini-Iamond Ring, you can use it to explore and solve all those. Or get your own copy of Tetrapentos, in tray or pouch.

A new kind of multi-polyform is Jacques Ferroul's La Ora Stelo, using combinations of the two isosceles triangles related to the golden ratio. Jacques has named them polyores. They've opened a whole world of new explorations. You can order them in the Pentagon Universe section.

Another new multi-polyform is Jacob Lettie's StarHex-II, with hexagons surrounded by equilateral triangles in combinations of up to 4 elements. We've named them polystars. You can see and order this set in the Tilings and Designs section.

Color-clad polyforms

Unusual relatives of polyforms are Kadon's Triangoes and Triangoes Jr., which combine the characteristics of both polytans and edgematching color sets, and Cubits, created by Anneke Treep and Christian Freeling, with shapes of one, two and three diamonds bearing colors that plot into a cube-like color tiling.

A cousin to Cubits is Diamond Star, where the diamonds' colors form stars instead of cubes.

In 2009 we introduced Christian Freeling's Q-Bix, whose polyhexes orders 1 through 3 are inlaid with 3 colors to form pure cube tilings. You can see and buy all these color-clad polyforms in the Tilings and Designs section.

New ideas are emerging continually. Stay tuned.

Our domain

Kadon is the world's pre-eminent producer of polyform puzzles for hands-on enjoyment and as a playable art form. In 2007 we procured the Internet domain name,, as a portal to our main website and to its polyforms and polyominoes sections.

To "Essential Polyforms" page

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