### 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.

### 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** **

### Hyperbola Clock

This amazing clock available here:

From Amazon: **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)

### Oloids: Solid and Anit-oloid

Order your Anti-Oliod today: available in three types of metal:

From KickStarter: **ORDER NOW: Anti-Oloids in Brass, Copper, and Steel **

Beautiful solid oloids available right now here:

From the Matter Collection:** BUY NOW: Oloids in Brass, Copper and Steel**

Oloids: “solid hull” and “ruled surface” types made from brass and copper- oloids are unique solids that roll in such a way that every point on their surface comes in contact with the plane. The basis of the oliod’s geometry is that of two connected circles, one perpendicular to the other such that the rim of each circle goes through the center of the other. The shapes you see here are the results of connecting the rims of these circles together with a family of straight lines, one method leads to the solid convex hull form, and another way leads to the ruled oloid (anti-oloid).

### Dudeney's Dissection

Get this version here:

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. 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.

### 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.

### Hyperbolic Holes

This inexpensive kit available here: From eBay: ** BUY NOW**

Hyperbolic Holes Kit

Hyperbolic Holes: a straight rod, in this case a pencil, glides through a symmetrical pair of curved holes. The design is based on the hyperboloid, the 3D ruled surface traced by an offset rotating diagonal line. This device is sold as an inexpensive kit to assemble yourself, and includes a motor with geared drive and pre-cut pieces. The pencil is my addition- just the right size to clear the curved openings.

### 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.

### 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!

### 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).

### Shashibo Geometric Art

**ORDER HERE : **Shashibo Geometric Art

Shashibo Geometric Art: dissect a cube into 12 equal irregular tetrahedra, connect these pieces symmetrically with hinges, and add 36 magnets to create a device with more that 70 geometrically interesting and aesthetic configurations- a few are shown here. With practice the various transitions can flow very smoothly- this video are some of my initial explorations with this unique puzzle.

### 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.

### Shashibo: Cube to Rhombic Docecahedron

**ORDER HERE : **Shashibo Geometric Art

Shashibo Transformations: a rhombic dodecahedron transforms into a cube- two possible configurations of this amazing dissection puzzle. The Shashibo is a cube cut into 12 equal irregular tetrahedra- these pieces are connected symmetrically with hinges, and 36 hidden magnets then allow more that 70 stable and geometrically interesting configurations to be discovered- swipe to see few more.