### 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.

### Pencil Hyperboloid

Choose your color and get one here:

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Hyperboloid Pencil Holder

don't forget a set of pencils:

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Colored Pencil Sets

Better yet- get some thermochromic color changing pencils!

From Educational Innovations: **BUY NOW **

Heat-Sensitive Pencils

Pencil Hyperboloid: a perfect gift for any math teacher- the precisely oriented holes in this base direct 16 pencils to reveal a hyperboloid, the 3D surface traced by revolving a diagonal(skew) line, the outline of which is the conic section of the hyperbola. A doubly ruled surface for any desktop!

### 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.

### 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.

### The Galton Board

Available here:

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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).

### 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.

### Half Seirpinski Octahedron Fractal

Get this amazing 3D print here:

From Etsy: **BUY NOW: Seirpinski Pyramid**

or print it yourself:

From Thingverse: Seirpinski Pyramid

Half Sierpinski Octahedron Fractal: this 3D printed math sculpture is one half of the sixth iteration of what is called “the octahedron flake” a 3D fractal based on the Sierpinski triangle. To make this fractal, on each iteration an inverted triangle is removed from the center of the previous triangle, and if this process is repeated indefinitely one gets the famous fractal. This 3D print used rainbow silk PLA to create the beautiful color gradient base on the .stl files by Rick Tu. Another example of math brought to life via 3D printing!

### 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.

### Aristotle's Wheel Paradox

Get this demonstration puzzle here:

From Etsy: **BUY NOW: Aristotle's Wheel**

WIkipedia has some details on the Wheel "Paradox"

Aristotle’s Wheel “Paradox”: How does the smaller attached disk travel the same length as the larger one if both disks only make one full rotation? Note the shorter path of the smaller disk, if rolled on its own. This beautifully made demonstration depicts an issue of geometry and motion that perplexed the best minds of humanity for 2000 years. The ancients knew the formula for circumference, and C=2πR for the large disk is clearly greater than C=2πr for the smaller- so how could the smaller disk, rotated once, still travel the distance of the larger one if attached? The great Galileo even offered a solution to the problem in his book Two New Sciences, where he approximated the situation as concentric hexagons and considered the limit as the number of sides increased. So what is the best answer to make sense of this situation?

### Penrose Aperiodic Rhombs

This set avaiable here:

From : **Search NOW Penrose P3 Tiling**

Wikipedia has the details on Penrose Tilings and their inventor Sir Roger Penrose (Recently won Nobel Prize!)

Another nice version of Penrose Tilings is available here:

From Etsy:** BUY NOW Penrose Tiles **

Penrose Aperiodic Rhombs: a famous aperiodic tiling with just two shapes- a pair of rhombuses with equal sides, but with the ratio of their areas made to equal the golden ratio. Note that although the starting pattern of 10 tiles is symmetrical, adding any further tiles breaks the symmetry, as highlighted by the path of the double curves. Sir Roger Penrose- who just won the Nobel prize in physics for his contributions to General Relativity- also discovered tessellations (tilings) that are aperiodic even though the two tile types are regularly shaped polygons. If one tries to shift a part of a Penrose tiling, the shifted part will not align or match up with any other part of the same tiling- all the way out to infinity! In this construction, single and double line patterns must align such that the tiles can only connect in specific ways to ensure the non-repetitive nature of the Penrose tiling structure. Shown here is one way these two tile types will fill the plane.

### Trammel of Archimedes

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Trammel of Archimedes

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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!

### 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!

### 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.