### Klein Bottle

Get this mathematical glass art here:

From Etsy: **BUY NOW: Klein Bottle**

Wikipedia has great details on the Klein Bottle

Klein Bottle: 3D representation of a four dimensional mathematical object with one side, no edges, and zero volume. Kind of like a MoĢbius strip with no edges.* Math meets glass art with this fun “lamp shaped” version! *only achievable in 4D space

### 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!

### Square Dissection Puzzles

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 Dissection Puzzles: a square can be cut (dissected) into polygons and then reassembled into other regular polygons. Shown here: an equilateral triangle, a pentagon, and a hexagon. These are the record holders for smallest number of pieces needed: triangle (4 pieces by Henry Dudeney 1902), hexagon (5 pieces Paul Busschop 1870s) and pentagon (6 pieces Robert Brodie 1891). Fun fact- It is not known if any of these records are the smallest possible, no mathematical proofs yet exist on this question.

### CMY Platonic Solids

Available from these sources:

From CMY Cubes: **All 5 Platonic Solids**

From Amazon: **BUY NOW: CMY Cubes and More**

Check out this nice set of Platonic Solids gaming dice

CMY Platonic Solids: The five famous convex regular polyhedra with thin film coatings, Cyan, Magenta, and Yellow, that allow filtering of the light that enters or leaves particular facets of each acrylic prism. I was struck by the intricate patterns produced in each object’s shadow by the refracted light- especially as the objects rotate.

### Logarithmic Spiral Gears

Amazing kinetitc creations made here:

From Etsy:** 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.

### Interactive Logic Gate Display

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From Etsy: **BUY NOW: Interactive Logic Gate Display**

Interactive Logic Gate Display: the most basic four binary/Boolean logic operations that are the fundamental components of modern computers- presented here appropriately on a standard green printed circuit board (PCB) complete with the truth table for each device. These logic gates are the basic building blocks of any logic circuit, from multiplexers, arithmetic logic units, and computer memory to full microprocessors which may contain hundreds of millions of such gates on a microchip. An educational, elegant, and fun design by Tyler Jacobs.

### Infinity Cube Sculpture

Get an infinity cube here (many colors and sizes to choose from):

From Ricardo Churchill (Etsy): **BUY NOW: Infinity Cube**

Infinity Cube Sculpture: crafted from solid stainless steel and powder coated orange, the geometry of this mathematical sculpture is the perimeter of the faces of a cube traced by a nonintersecting connection of equal line segments. When viewed from a corner, a cube has a hexagonal cross section, and some may recall the famous logo of Silicon Graphics computers based on such an infinity cube (a design by Scott Kim). Rotating this “tubed cube” along a diagonal axis reveals the interesting symmetries of this geometric construction.

### Spirograph Animator

Get this amazing and inexpensive toy here:

From Amazon: **BUY NOW: Spirograph Animator**

Spirograph Animator: a turntable adds an ingenious embellishment to this classic math toy. When the centerpiece is put into rotation a strobe light turns on and the cyclic designs spring into a complex dance. Spirographs produce mathematical roulette curves called hypotrochoids, cyclical designs that repeat at varying integer multiples. When the strobe light frequency, and rotation frequency, match up with an integer multiple of one of the patterns, that roulette curve appears stationary while the other patterns stay in motion. Shown here is but one combination of many possible overlapping patterns that can be produced by the supplied set of gears and colored pens.

### Sphericon Frame Roller

Get one here:

From Etsy:** BUY NOW: Quantum Sphericon**

Sphericon Frame Roller: beautiful 3D printed version of a developable roller (an object that rolls where every point on the roller’s surface comes into contact with the plane upon which it rolls). The sphericon (based on a square) rolls in a straight line with a peculiar wobble motion and was discovered in 1980 by David Hirsch and is but one of a family of such rollers called polycons.

### 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.

### The Galton Board

Available here:

From Amazon: **BUY NOW **

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).

### Tessellating Geckos

Laser cut 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.

### Scutoids

This set of 3D printed scutoids available here:

From Etsy: **BUY NOW: Scutoid container set**

Scutoids: a recently recognized form of geometric solids discovered in 2018. Scutoids have a different polygon on each end, and when packed together maximize stability and minimize energy when forming a boundary layer- such as the membranes around organs in living tissue. These 5-6 scutoids have pentagons on one end and hexagons on the other. Discovered by a collaboration of biologists and mathematicians in 2018, this set is sold by Recep Mutlu of 3DPrintBase.

### 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.

### Trammel of Archimedes

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From Etsy:** BUY NOW **

Trammel of Archimedes

From eBay:** BUY NOW **

Trammel of Archimedes

Trammel of Archimedes: as the shuttles take turns completing their straight line journeys, the end of the crank arm traces an ellipse. Sometimes sold as a “do nothing machine” or “nothing grinder”, far from doing nothing this simple and crucially important mechanism demonstrates how rotational motion can be converted into translational oscillatory motion- such as how a piston can drive an engine’s crankshaft. This version was crafted from fine maple, cherry, and oak by artisan Neal Olsen.

### Soma Cube

The Soma Cube is available in a variety of materials and colors:

From Amazon: **BUY NOW: Soma Cube**

From Etsy: **BUY NOW: Soma Cube**

Soma Cube: Math toy invented by Danish scientist and artist Piet Hein, who claimed that this puzzle idea came to him as he was listening to a lecture on Quantum Mechanics by Werner Heisenberg (yes- that Heisenberg) in 1933. The seven pieces are all the ways 3 or 4 cubes can be joined, such that each piece has at least one inside corner. Amazingly there are 240 ways to make the larger cube from these 7 pieces- still not that easy!