### 120 Sided Fair Dice

Get one here! Many colors to choose from.

From Amazon:** BUY NOW 120 sided dice **

120 Sided Fair Dice: mathematically this die has the maximum possible number of sides with equal area. Two mathematicians, Robert Fathauer and Henry Segerman, realized that the oddly named regular 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!

### 3D Pentominoes

The set I used for this video is called Pocket Katamino and is available here

From Amazon:** BUY NOW Pentominoes **

3D Pentominoes: the 12 possible arrangements of five identical squares, joined edge to edge, form the set of all pentominoes. Since 12x5=60, the pentominoes can tile a 6 x 10 rectangle with no gaps (2339 ways to do this- yet even finding one solution is a challenge). This set of colorful pentominoes is made so that the height of each piece is the same as the width of the constituent squares, such that 3D constructions can be made. Since 3x4x5=60 one can build a box with these dimensions (amazingly, 3940 ways to do this- but again, finding one is still a fun challenge).

### Square Kaleidocycle

This book has many versions of kaliedocycles: cut out and glue to make many interesting mathematical objects.

From Amazon: **BUY NOW: MC Escher Kaleidocyles**

Square Kaleidocycle: a ring of eight linked tetrahedra. The hinged connections allow the ring to be rotated through its center. The faces of the pyramids are decorated with the famous tessellation work of MC Escher, a pattern of interlocking lizards. Note that as the kaleidocycle is rotated the lizards at the center change through each of four colors. Made from card stock, this kaleidocycle was cut and assembled from a book by mathematicians Doris Schattschneider and Wallace Walker.

### Beaded Kaleidocycle

Get similar beadwork geometric art here:

From Etsy: **BUY NOW: Beadwork Kaleidocycle**

Beaded Kaleidocycle: based on a geometry of six linked tetrahedra with hinged connections that allow the ring to be rotated through its center. Intricate beadwork meets math in this kinetic artwork by Erin Peña.

### Mirror Illusion Motion Bank

A few of these are available on eBay as of this posting.

From eBay: **BUY NOW: Mirror Illusion Bank**

Mirror Illusion Spacecraft Bank: Symmetry + Reflection = Illusion. Deposit a coin (which seems to vanish) and a spacecraft of some sort states to revolve within a silver ring seemingly suspended in space. The symmetry of the ring and craft allows a half of these objects to appear as a whole, and a AA battery powers a motor which drives gears to slowly spin the portion of the mirror inside the ring (with the spacecraft attached). The best version of the mirror illusion box I’ve seen.

### 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 ?

### 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.

### Skew Dice

Available here!

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.

### 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!

### Pocket Scintillator Kinetic Art

Logan sometimes has items for sale here:

From Etsy: **BUY NOW: PocketScintillators**

Pocket Scintillator Card: three sheets of seemingly random arrays of translucent colored pixels produce words and images when stacked- shift the stack of sheets and a second images appears! Innovative kinetic optical art by inventor, artist, software developer Logan Kerby @thanksplease who kindly sent me these cards encoded with @physicsfun themes.

### Hyperbola Clock

This amazing clock available here:

From Maths Gear: **BUY NOW **

Hyperbola Clock

Hyperbola Clock: a straight rod glides through a curved hole in this unconventional clock based on the hyperboloid, the 3D ruled surface traced by rotating diagonal line. In this creation by Robert Darwen of Fibonacci Clocks, the rod serves as the hour hand with a smaller minute hand above the center of the base disk. (The time adjustment dial of the clock mechanism was connected to a small motor to produce the sped up motion in this video so that 1 second = 1 hour)

### Steinmetz Bicylinder

Three choices of metal- order one today!

From KickStarter: **ORDER NOW: Steinmetz Bicylinder**

Steinmetz Bicylinder: intersect two cylinders at right angles and the remaining confined space is the bicylinder- shown here machined from stainless steel. The bicylinder casts a circular shadow along two orientations, and a square shadow perpendicular to those. In addition the curve created along where the two cylinders meet is an ellipse- as seen with the object spinning along the intersection axis. Fun fact: the area and volume of this object are known to be A=16r^2 and V=16r^3/3. Thanks to Zac Eichelberger of Math Meets Machine for sending me one of his creations.

### Tessellation Origami

See more of Ekaterina's amazing work on her website gallery:** Kusudama me! **

Contact her to buy her artwork, or you can buy her books and learn how to fold amazing geometries!

From Amazon: **BUY NOW: Ekaterina Lukasheva: Papercraft and Origami**

Tessellation Origami: nested spirals and triangles created from one flat sheet of paper! This beautiful work by Ekaterina Lukasheva also demonstrates how folded paper can obtain very different physical properties than that of the original flat paper. When stretched out this paper sculpture prefers to snap back into spirals and triangles, and although most materials bulge out when compressed along one direction, here the design compresses evenly along all three axis of the hexagonal symmetry.

### The Holoscope: Cube with Spheres

Order a holoscope from the artist's gallery here:

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

The Holoscope: a cube of mirrors with the interior viewed from one corner and illuminated by light entering from glass spheres at the other seven vertices. A type of kaleidoscope based on truncated Platonic solids by artist Gary Allison. Each holoscope has stained glass on the exterior and front surface mirrors on the inside which create the amazing and seemingly impossible spaces within.

### Ambiguous Object

Available here:

From Amazon (Japan):** BUY NOW set of four ambiguous objects with booklet **

This kit contains four white plastic illusion objects (including the object in the video) and a booklet. I used the translate feature in the Chrome browser to place my order and it shipped to California in three days.

Some 3D prints are available from many makers here:

From eBay:** BUY NOW Ambiguous Objects **

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 **

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.

### In and Out Illusion

A similar arrow illusion available here:

From Etsy:** Buy Now: Stubborn Arrow Illusion **

In or Out Illusion: this 3D printed sculpture incorporates the now famous Stubborn Arrow Illusion and features both a left and right handed version. These ambiguous object illusions are a fairly recent invention by mathematician Kokichi Sugihara of Meiji University in Japan which take advantage of a clever combination of perspective, and viewing angle.

### 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.

### 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.