Math Toys

Pythagorean Puzzle

Available here: 
From Creative Crafthouse: BUY NOW Pythagorean Puzzle

Pythagorean Puzzle: a proof, in physical form, of one of the most famous equations concerning the sides of any right triangle. The area of a square with side c of the hypotenuse is indeed equal to the sum of the areas of the squares of side a and b. This kit also allows at least two other ways to prove this theorem named after the famous Greek mathematician from 500 BC. One of the most used formulas when calculating vectors in physics classes ?

The Holoscope : Icosahedron

Order a holoscope from the artist's gallery here: 
The artwork of Gary Allison: BUY NOW 

Look through other holoscopes in my collection here: Holoscope Kaleidoscopes

The Holoscope Icosahedron: the intricate beauty of multiple internal reflections from 20 triangular mirrors in the shape of this famous platonic solid. The interior is viewed from one corner and illuminated by light entering from glass spheres placed at all of the other 11 vertices. A type of kaleidoscope based on mirrored polyhedra by artist Gary Allison, (swipe to see the dodecahedron and cube) and future posts will include tetrahedron and octahedron holoscope forms as I complete my Platonic solids set. Each holoscope has stained glass on the exterior and front surface mirrors on the inside which create the amazing and seemingly impossible spaces within.

Sphere Sticks Geometric Puzzle

Get this affordable and amazing puzzle here:

From Etsy: BUY NOW: Sphere Sticks

Sphere Sticks Puzzle: 30 identical wood pieces, each with two notches as shown, can create 12 interlocking pentagons in a perfect symmetry- look carefully and you can see that each rod is in an identical configuration with the 4 others that connect with it. Precision cut notches on the rods allow them to interlock with elastic tension such that vector sum of the 4 forces sum to zero in this tensegrity type equilibrium. The dodecahedron, with its 30 edges and 12 sides, is the basis of this puzzle sculpture. 


Rhombic Blocks Mathematical Puzzle

This beautifully made puzzle available here:: 
From Etsy: BUY NOW 
Rhombic Blocks 

Rhombic Blocks Mathematical Puzzle: There are 9 possible ways three rhombuses can be joined together along a common edge, and similar to pentominoes, these 9 tri-rhombs can tile a polyhedron, in this case a hexagon. There are 14 solutions to this puzzle, and one where no same-colored pieces touch. A beautiful math discovery by puzzle master Stuart Coffin.

Tension Integrity Icosohedron: Tensegrity

A nice version of this tensegrity icosohedron is sold as a toy for tiny tots: 
From Amazon: BUY NOW Tensegrity Toy 

I constructed this version by referring to the images and descriptions of tensegity on Wikipedia 

Tension Integrity Icosohedron: Six brass struts float isolated from each other but held in a stable configuration by a net of 24 connecting cables. I made this sculpture using hollow brass tubes and weaving through them a single strand of fishing line, which is connected after passing through each tube exactly four times. This configuration of three sets of parallel struts forms a Jessen’s icosahedron under tension, and was invented by the famous architect Buckminster Fuller in 1949.

The DeltaCELT Rattleback 

Get one here: 

From Etsy: BUY NOW: The DeltaCELT

The DeltaCELT: a new rattleback design where the indentations create an asymmetry in the distribution of mass that leads to rotation reversals when spun: 1) If spun clockwise, after about two full rotations, a complicated combination of friction, precession, and instability induced vibrations transforms the rotational energy into into end-to-end rattling (energy of oscillations) and then into rotational energy in the opposite direction. 2) If spun counter-clockwise, after about 30 full rotations the rotational energy translates into side-to-side oscillations leading to a clockwise reversal (swipe to see- less dramatic but still amazing). The design is made of solid brass and has the shape of a bisected prolate ellipsoid in the proportion of the mathematical constant δ =4.669.. (Feigenbaum dynamical constant) giving the name to this new design by astrophysicist Kenneth Brecher. The DeltaCELT comes with the polished slate spinning surface for optimal performance. 

Anamorphic Harry Potter Puzzle

No longer in production but available on eBay: 

From eBay: BUY NOW: Harry Potter On Reflection Puzzle

Mirror Anamorphic Harry Potter: the conical mirror of the silver container reflects and reveals the distorted scenes in this 200 piece jigsaw puzzle of the famous wizard and school. The shape of the mirror allows for a mathematical operation, a type of affine transformation, to map the distorted image of the puzzle to the restored image reflected by the mirror. Real magic! 

Nova Plexus Interlocking Puzzle

Precision machined and available in brass or stainless steel:

From Art of Play: BUY NOW: Nova Plexus Puzzle

Nova Plexus Puzzle: 12 identical brass rods can create 4 interlocking triangles in a perfect symmetry- look carefully and you can see that each rod is in an identical configuration with the 5 others that connect with it. Precision machined notches on the ends of the rods allow them to interlock with elastic tension such that vector sum of the 5 forces on each rod is zero- creating this astonishing geometry as the equilibrium state. Unlock the ends of any two rods and the system instantly disassembles (swipe to view process in slow motion). Invented/designed by artist and computer scientist Geoff Wyvill in 1978, this puzzle has just recently been made available for sale with a limited production run.

Skew Dice

Available here! 
From Amazon: BUY NOW Skew Dice

Skew Dice: these unusually shaped dice are completely fair- roll them and the probability of outcomes are identical to a standard set of dice! The odd shapes are a special type of polyhedra called asymmetric trigonal trapezohedra which come in right and left handed versions- this set has one of each (mirror images of each other). What allows this shape to be fair like a cube has to do with the property of being isohedral, where each face of an object will map onto all other faces via a symmetry of the object. Manufactured by The Dice Lab. 

Ambiguous Object Illusion Mug

From Etsy: BUY NOW Squirkle Mug Ambiguous Object Illusion 

These type of objects were invented by mathematician Kokichi Sugihara, and you can buy his books here: 
From Amazon: BUY NOW Ambiguous Objects by Kokichi Sugihara 
The math and physics are described here in this technical journal article by Prof. Sugihara

Ambiguous Object Illusion Mug: circle or a square? It’s all a matter of perspective and viewing angle. The complex shape allows for both to be perceived and is based on the work of mathematician Kokichi Sugihara of Meiji University in Japan, the inventor of this illusion and art form. 

High Voltage Wood Burn Fractal

Beautiful wood fractal art available here:

From Etsy: BUY NOW: High Voltage Fractal Art

High Voltage Wood Burn Fractal: a classic Lichtenberg fractal created by electrical discharge- in this piece artist/maker Nic Ferretti applied cobalt blue epoxy resin to fill the cavities of the burn pattern and create the look of frozen lightning. Swipe to see my attempt at this process with an old neon sign high voltage transformer using 7500 V at 20 mA. I turned the power off right before the two sides connected to get the fern-like look. The wood is slightly wetted with baking soda dissolved in tap water. WARNING high voltage can kill- do not use a transformer like this unless you are fully trained in electrical high voltage safety. 

Dodecagon to Square Puzzle

Get these and other well made disection puzzles here:

From Etsy: BUY NOW: Square Dissection Puzzles

These puzzles are expertly laser cut and sold by GamesEfce. I spray-painted the pieces of mine to better show the shapes and relationships for the video. 

Square to Dodecagon Dissection Puzzle: a square can be cut (dissected) into polygons and then reassembled into other regular polygons. Amazingly the 12-sided regular polygon can be cut into these six specific pieces that can then fit a square. Swipe to see the minimum cuts dissections for a hexagon, pentagon, and triangle.


Ambiguous Object 

These type of objects were invented by mathematician Kokichi Sugihara, and you can buy his other books here: 
From Amazon: BUY NOW Ambiguous Objects by Kokichi Sugihara 

Also available from Amazon (Japan): BUY NOW set of four ambiguous objects with booklet 

Similar objects available here- from Etsy: BUY NOW Ambiguous Objects

Another illusion design by Kokichi Sugihara of Meiji University in Japan, the inventor of this illusion and art form. A mathematically calculated combination of perspective and the physics of reflection produce this striking illusion that works in many configurations.

The Random Walker

Galton Board version available here: 
From Amazon: BUY NOW 
Galton Board 

The Random Walker: second model of two Galton Boards designed and produced by this version is made to demonstrate probability in investment returns of a global stock market portfolio relating to risk capacity. Slow motion reveals the erratic path of each steel ball (second half of video). The red graph shows the distribution of 592 monthly returns (mean =1%, SD=5%) representing data from 50 years of an IFA Index fund- here the random “walk” of 3000 steel balls falling through 12 levels of branching paths always produce a close match, and both distributions tend toward the famous bell curve distribution. A wonderfully designed modern version of the Galton Box invented by Sir Francis Galton(1894) to demonstrate the Central Limit Theorem - showing how random processes gather around the mean.

Hyperbolic Holes

This inexpensive kit available here:

From Amazon: BUY NOW: Hyperbolic Holes Kit

Hyperbolic Holes: a straight rod, in this case a pencil, glides through a symmetrical pair of curved holes. The design is based on the hyperboloid, the 3D ruled surface traced by an offset rotating diagonal line. This device is sold as an inexpensive kit to assemble yourself, and includes a motor with geared drive and pre-cut pieces. The pencil is my addition- sharpened to just the right size to clear the curved openings.

Lissajous Roller 

Available from Pyrigan & Co.

From Etsy: BUY NOW: Lissajous Roller Illusion

Lissajous Roller: when viewing this 3D printed object from the side one sees a projection of a 3:2 Lissajous curve, but the object is actually cylindrical in frame and can roll towards or away from the viewer. When in motion a “dual axis illusion” is produced where the object appears to be rotating about a vertical axis. Invented by Bill Gosper and produced by Pyrigan & Co. 

10 Hex Puzzle

This and other beautiful and well made puzzles are available on Etsy: 
From Etsy: BUY NOW 10 Hex Puzzle 

Two great resources about these polyhexs: polyform puzzler page and puzzleworld page 

10 Hex Puzzle: this puzzle is comprised of pieces which are the set of all ways three and four hexagons can be joined with a common edge. There are 3 trihexs and 7 possible tetrahexs, and similar to pentominoes, these 10 polyhexs can assemble into a large hexagon. Amazingly there are exactly 12,290 solutions to this puzzle- but it’s still a challenge to find just one!