### The iTOP

Well engineered brass spinning toys by SiriusEnigmas:

From Etsy: **BUY NOW: The iTOP **

From Etsy: **BUY NOW: The PhiTOP**

From Etsy: **BUY NOW: The eTOP**

From Etsy: **BUY NOW: The PiTOP**

The iTOP: inverting spinning disk- the equilibrium state for a spinning thing is often different from that of the same object when stationary. When spun, the iTOP almost instantly inverts to raise its center of mass (shown in slow motion because it happens so fast). However, when the rotation speed decreases to a certain rate the system becomes unstable (shown again in slow motion) and flips again before going into a rolling/spinning motion like a coin. This top completes the set of four spin tops by astrophysicist Kenneth Brecher, all made of polished brass and themed on a mathematical constant. Swipe to see a demonstration of each: iTOP (square-root of -1), PhiTOP (golden ratio, φ), eTOP (base of the natural log), and the PiTOP (C/D of a circle, π).

### Hexacon and Sphericon Rollers

Get these and other amazing developable rollers here:

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.

### 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).

### 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. For this 3D print, the maker Travis Quesenberry 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!

### 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.

### Arrow on Mobius Strip

Get this 3D printed object of math topology here:

From Etsy: **BUY NOW: Mobius Strip with Arrow**

Arrow on Möbius Strip: on the geometry of a Möbius strip a right pointing arrow points left after one trip around, a second trip restores the original orientation. This mathematical property is called non-orientability, and is also true of Klein bottles which I’ve posted about. I love how this 3D printed model, designed and produced by Wes Pegden, allows one to physically manipulate and intuit this somewhat obscure mathematical property.

### 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!

### The PiTOP

Well engineered brass spinning toys by SiriusEnigmas

From Etsy: **BUY NOW: The PiTOP**

From Etsy: **BUY NOW: The PhiTOP **

The PiTOP: this beautiful brass cylinder has a radius of 1 inch, a height of 1/π inches, and displays the first 109 digits of π on its face. Spin this disk (best with sound on) and it will demonstrate some very interesting physics involving energy transfer and conservation of angular momentum. The edges of the cylinder are rounded and engineered to exhibit an optimum motion of a “spolling” coin, a motion that combines spinning and rolling (closeup shown in 240fps). As the disk’s angle of inclination decreases the speed of the rolling increases dramatically until the contact point with the mirror is moving in excess of 200mph. From the mind of physicist Ken Brecher, inventor of the PhiTOP (swipe for video). Note also that with the specified height, that the volume of this cylinder is exactly 1 cubic inch!

### Reuleaux Rotor

This toy is availble from Amazon Japan and will ship to the US:

From Amazon.jp: **BUY NOW: Reuleaux Rotor Wodden Toy**

Reuleaux Rotor: this famous curve of constant width, the Reuleaux triangle, can rotate such that at all times it remains in contact with all four sides of a square. As demonstrated by this wooden toy from Japan, the rotor covers approximately 98.77% of the area of the square, missing only the sharp corners. The curves in the corners are in the shape of an elliptical arc. Fun fact: a Reuleaux triangle has a perimeter equal to pi times its width- just like a circle!

### RPSLK Dice

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.

### 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.

### Orbiforms

Latest orbiforms available here:

From Kickstarter: **Order NOW **

Orbiforms in Steel, Brass, or Copper

Orbiforms: volumes of constant width made from solid steel, brass, and copper- these shapes have constant diameter no matter their orientation and will roll like spheres between two planes- note how the acrylic plate stays parallel to the table as the orbiforms roll underneath. The first set shown are based on the Reuleaux triangle and the second set are based on a Reuleaux pentagon. Currently available on Kickstarter from my friends at @altdynamic** **

### Stomachion Puzzle

Get this 3-color laser cut acrylic version here:

From Kadon Enterprises: **BUY NOW: Stomachion Puzzle**

Also a very nice multicolor acyrlic version here:

From Etsy: **BUY NOW: Stomachion Puzzle**

Learn about the 1998 discovery of the lost writings of Archimedes (and the technology used to recover them) in this TED talk.

Ancient Stomachion Puzzle: the oldest known puzzle, discovered in the writings of the great Greek physicist and mathematician Archimedes from some 2200 years ago. The puzzle is a dissection of a square into 14 polygons, where the areas of each piece are integer multiples of each other (a curious way to slice it up). In 2003 Bill Cutler showed that there are 536 district ways to configure these pieces to make the square (five are shown here), ignoring simple rotations and reflections. Swipe to see the most famous solution, attributed to Archimedes himself, that was found in an ancient manuscript discovered only in 1998- before this date historians knew the name of the puzzle, but no one knew what it looked like. Kate Jones, the maker of this particularly aesthetic version, found that when using only three colors for the polygons, there are only 6 solutions where no two pieces of the same color touch (four solutions shown here).

### 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.

### Holoscopes: Dodecahedron and Cube

Order a holoscope from the artist's gallery here:

The artwork of Gary Allison: **BUY NOW Holoscopeworld.com **

Look through other holoscopes in my collection here: Holoscope Kaleidoscopes

Holoscopes: polyhedra of mirrors (dodecahedron and cube) with the interior viewed from one corner and illuminated by light entering from glass spheres placed at all of the other vertices. A type of kaleidoscope based on mirrored polyhedra by artist Gary Allison @holoscope2000, and future posts will include tetrahedron, octahedron, and icosahedron 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.

### Superellipsoid

Vintage Super Eggs of Piet Hein can be found on eBay:

From eBay: **BUY NOW **

Superellipsoid by Piet Hein

Get one in brass from this shop:

From Etsy:** BUY NOW **

Brass Super Egg

Superellipsoid: the "super egg" is a mathematical creation of Danish scientist and artist Piet Hein (also the inventor of the Soma cube puzzle). The equation of the superellipse is that of a regular ellipse, but raising both sides to the power of 2.5 instead of 2- the resulting curve had a flattened end that allows the superellipsoid to stand upright. The superelljpse has found use in architecture and design. This super egg is made of stainless steel.