Math Toys

Hexacon and Sphericon Rollers

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From Etsy: BUY NOW: Hexacon and Sphericon Rollers

Hexacon Roller: beautiful 3D printed versions of a recent mathematical discovery of new developable rollers (objects that roll where every point on the roller’s surface comes into contact with the plane upon which it rolls). Similar to the sphericon (based on a square) the hexacon rolls in a straight line with a peculiar wobble motion but has a hexagonal cross section (swipe to see video loop of each in motion). The hexacon (2019) and sphericon (1980) are two of a family of such rollers called polycons discovered by David Hirsch, and described in a paper by Hirsch and Seaton published in January of this year. 

Hyperboloid Spinner

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From Amazon: BUY NOW 
Hyperboliod Spinner: The HypnoGizmo 

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Hyperboloid Spinner: HypnoGizmo 

Hyperboloid Spinner: the HypnoGizmo toy consists of a set of slanted straight nylon lines arranged to form the outline of a hyperboliod- the quadratic surface related to the revolution of hyperbola around its axis of symmetry. As the device rotates the beads slide along in succession on one of the straight paths leading to the complex visual display. So much fun math in this toy!

Skew Dice

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From STEMcell Science: 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. 

120 Sided Fair Dice

Get one here! Many colors to choose from. 
From Amazon: BUY NOW 120 sided dice 

he d120: mathematically this die has the maximum possible number of sides with equal area (discovered so far). Two mathematicians, Robert Fathauer and Henry Segerman, realized that the oddly named uniform convex polyhedron (disdyakis triacontahedron) had the needed geometry to make a 120 sided fair die. Like the familiar 6 sided die, the d120 has the following properties: every side must have equal area and the numbers on parallel sides (top and bottom) must sum to the same number. The inventors admit that they do not have any suggested use for these dice- they made them purely because mathematically it was possible to do so! 

The Galton Board

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Galton Board 

The Galton Board: 3000 steel balls fall through 12 levels of branching paths and always end up matching a bell curve distribution. Each ball has a 50/50 chance of following each branch such that the balls are distributed at the bottom by the mathematical binomial distribution. One of my favorite finds of 2018! An elegantly 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. In addition the number of balls in each bin can be predicted by Pascal's triangle (printed on the face over the pegs).

Dudeney's Dissection

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From Grand Illusions Ltd: Dudeney's Dissection 

A nice wood version is available here: 
From Etsy: BUY NOW Dudeney's Dissection 

See both Wikipedia and Wolfram MathWorld for more details on the history and math of this geometrical oddity. 

Dudeney's Dissection: an equilateral triangle can be cut (dissected) into four pieces that will then assemble into a square. Interestingly the four parts are all different in shape (the green and yellow pieces are similar but not the same). This hinged model is comprised of precision machined and anodized aluminum, and can be folded back and forth between the two simplest regular polygons. It is not known if a similar three piece dissection is possible. Also called the haberdasher's problem and described in 1907 by Henry Dudeney it is the only 4 piece solution known.

Pythagorean Puzzle

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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 ?

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! 


These dice are available here:

From Amazon: BUY NOW: Rock Paper Scissors Lizard Spock Dice

RPSLK Dice: the famous Lizard Spock extension to the Rock, Paper, Scissors game expressed on 10 sided dice allowing the study of the non-associative nature of the game (Rock wins Scissors, and Scissors wins Paper, but Rock does not win Paper, etc.), and other interesting math. The original RPS game had three “weapons” and only three rules are needed to play the game. Adding Lizard-Spock makes for 5 gestures, but now 10 rules must be used, including “Spock vaporizes Rock”, “Lizard poisons Spock”, and my favorite “Paper disproves Spock” (swipe to see famous graphic). Interestingly, mathematical analysis shows a similar four weapon game with equal odds of winning is not possible. It was also found that the next possible game with 7 gestures would require 21 rules to play. The Lizard-Spock extension was invented by Sam Kass and Karen Bryla in 2005 and made famous on the sitcom Big Bang Theory. 

In-Feed Google

Tessellating Geckos

These laser cut hardwood geckos are available here: 
From Etsy: BUY NOW Tessellating Geckos 

Tessellating Geckos: MC Escher inspired lizard cutouts interlock precisely to tile a surface with no overlaps or gaps. Laser cut from maple, walnut, and cherry wood by maker/artist Craig Caesar and inspired by MC Escher’s “Study of Regular Division of a Plane with Reptiles” 1939. G4G week: Martin Gardner wrote about the art and math of Escher in 1961- which helped create the popularity that his work has experienced ever since.

Hexa Sphericon

Sphericon and Hexa-sphericon: order your set today! 
From the Matter Collection: BUY NOW The Sphericon (Hex and Regular) 

Hexa-Sphericon: Sphericons are unique solids that roll in such a way that every point on their surface comes in contact with the plane. Solids from the sphericon family all have one side and two edges. Each sphericon is based on a regular polygon, with the basic sphericon derived from a square, and here- a more interesting case with more complex rolling motion- from a hexagon. 

Logarithmic Spiral Gears

Amazing kinetitc creations made here: 
From Etsy: BUY NOW 
Spiral Gear Set  and Motorized Sprial Gears

3D Print avialble here:
From eBay: BUY NOW: Spiral Gear Set 3D Printed

Original 3D print files available here: 
From Thingverse: Spiral Gear Set 

Logarithmic Spiral Gears: an extreme example of non-circular gear sets. This set is based on the famous Fibonacci spiral and evokes the cross section of nautilus shell with internal chambers. If one gear of this set is turned at constant speed, the other will turn with an varying speed. A laser cut based on 3D prints of Misha Tikh and the research of Balint et al. 

The Klein Bottle

The best Klein Bottles are made by Cliff Stoll, astronomer, mathematician and artist. Every one-sided, zero volume bottle is packaged and shipped by Cliff himself. Get one today! 
From ACME Klein Bottles: Buy NOW Klein Bottles by Cliff Stoll 

Wikipedia has great details on the Klien Bottle, and the amazing Cliff Stoll

The Klein Bottle: 3D representation of a four dimensional mathematical object with one side, no edges, and zero volume. Kind of like a Möbius strip with no edges.* Math meets glass art! Many thanks to Cliff Stoll for this kind gift and a great visit including a wonderful tour of his collection of mathematical oddities. *only achievable in 4D. 

Spherical Dice

A must for any die/dice collectors: 
From Amazon: BUY NOW Spherical Dice 

Click this link for other amazing dice featured on @physicsfun 

Spherical Dice: these fair six "sided" dice are hollow inside with a ball that weights each sphere such that one of the six values is always on top. When these dice are rolled (literally!) the internal weight lands in one of six cavities inside creating a low center of mass which aligns one of the numbers to the top. Another low center of mass toy! 

Sphere Sticks Geometric Puzzle

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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. 


Wooden Trapped Sphere in Cube

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From Art of Play: BUY NOW: The O-Cube

Trapped Sphere in Cube: a surprising aspect of the geometry of spheres and cubes- carved from a single block of wood is a sphere trapped within a cube frame. A classic folk woodworking novelty reconceived here with precision machining to create this seemingly impossible object.