### Ambiguous Object Illusion Mug

From Etsy:** BUY NOW Squirkle Mug Ambiguous Object Illusion **

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 **

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

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

The math and physics are described here in this technical journal article by Prof. Sugihara.

Ambiguous Object Illusion Mug: circle or a square? It’s all a matter of perspective and viewing angle. The complex shape allows for both to be perceived and is based on the work of mathematician Kokichi Sugihara of Meiji University in Japan, the inventor of this illusion and art form.

### 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!

### Tension Integrity Icosohedron: Tensegrity

A nice version of this tensegrity icosohedron is sold as a toy for tiny tots:

From Amazon:** BUY NOW Tensegrity Toy **

I constructed this version by referring to the images and descriptions of tensegity on Wikipedia

Tension Integrity Icosohedron: Six brass struts float isolated from each other but held in a stable configuration by a net of 24 connecting cables. I made this sculpture using hollow brass tubes and weaving through them a single strand of fishing line, which is connected after passing through each tube exactly four times. This configuration of three sets of parallel struts forms a Jessen’s icosahedron under tension, and was invented by the famous architect Buckminster Fuller in 1949.

### Nova Plexus Tensegrity Puzzle

Precisions machined and available in brass or stainless steel:

From Art of Play: **BUY NOW: Nova Plexus Puzzle**

Nova Plexus Puzzle: 12 identical brass rods can create 4 interlocking triangles in a perfect symmetry- look carefully and you can see that each rod is in an identical configuration with the 5 others that connect with it. Precision machined notches on the ends of the rods allow them to interlock with elastic tension such that vector sum of the 5 forces on each rod is zero- creating this astonishing geometry as the equilibrium state. Unlock the ends of any two rods and the system instantly disassembles (swipe to view process in slow motion). Invented/designed by artist and computer scientist Geoff Wyvill in 1978, this puzzle has just recently been made available for sale with a limited production run.

### Oloids: Solid and Anit-oloid

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

From The Matter Collection: **ORDER NOW: Anti-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).

### Lissajous Roller

Available from Pyrigan & Co.

From Etsy: **BUY NOW: Lissajous Roller Illusion**

From ShapeWays: **BUY NOW: LIssajous Roller Illusion**

Lissajous Roller: when viewing this 3D printed object from the side one sees a projection of a 3:2 Lissajous curve, but the object is actually cylindrical in frame and can roll towards or away from the viewer. When in motion a “dual axis illusion” is produced where the object appears to be rotating about a vertical axis. Invented by Bill Gosper and produced by Pyrigan & Co.

### Kinetic Traced Hyperboloid

This hard to find sculpture curretnly available here:

From Amazon: Hyperbolic Kinetic Sculpture

Kinetic Traced Hyperboloid: a straight rod glides through a symmetric pair of curved holes in this kinetic sculpture based on the hyperboloid, the 3D ruled surface traced by an offset revolved straight line. This version is made of anodized aluminum and rotates via gearing and a motor powered by two AA batteries in the base.

### 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).

### 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!

### 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.

### Cone of Apollonius

Similar models available here:

From Amazon: **BUY NOW: Cone of Apollonius**

From Etsy: **BUY NOW: Cone of Apollonius**

Cone of Apollonius: Slicing a cone with a plane will produce the famous curves known as the conic sections, as demonstrated with this beautiful vintage wood model by Nasco. Slicing at a right angle to the cone’s axis of symmetry produces a circle, and tilting the intersecting plane a bit produces an ellipse. When the plane is tilted parallel to the side of the cone the curve produced is a parabola, and tilting even further creates a hyperbola. The discovery of the mathematics demonstrated here are attributed to Apollonius of Perga from about 250 BC- thousands of years later Kepler, Newton, and others showed these conic sections to be intricately connected to many branches of physics such as planetary orbits and the optics of telescopes.

### Uphill Roller

This set available here:

From Amazon: **BUY NOW: Uphill Roller Double Cone**

Uphill Roller: a double cone (like two funnels connect by their tops) will roll up a set of inclined rails. Although the bi-cone rolls toward the higher end, its center of mass descends due to the geometry of the rails. This curious construction was first published in 1694 by the noted surveyor William Leybourn to promote “recreation of diverse kinds” towards the “sublime sciences”. Physics fun from three centuries ago!

### 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.

### Ambiguous Object

Available here:

From eBay:** BUY NOW set of four ambiguous objects with booklet **

This offical kit/book from Prof. Sugihara contains four white plastic illusion objects (including the object in the video) and a booklet.

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 other books here:

From Amazon:** BUY NOW Ambiguous Objects by Kokichi Sugihara **

Also available from Amazon (Japan):** BUY NOW set of four ambiguous objects with booklet **

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.

### Pencil Hyperboloid

Choose your color and get one here:

From Etsy: **BUY NOW **

Hyperboloid Pencil Holder

don't forget a set of pencils:

From Amazon: **BUY NOW **

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!

### Sphere and other Orbiforms

These volumes of constant width available for order now: choose from brass, copper, or stainless steel

From AltDynamic: **BUY NOW: Sphere and Orbiforms**

Sphere and other Orbiforms: pi day special post- volumes of constant width made from solid brass. 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 sphere and other orbiforms roll underneath. The first orbiform is based on the Reuleaux triangle and the second on a Reuleaux pentagon. Fun pi fact- the perimeter of any shape of constant width is alway equal to the diameter(width) multiplied by pi: P=πd.

### Dandelin Spheres

I found this beautiful model on eBay and I'm not sure of its age or origin.

Grant Sanderson describes the elegant geometry behind this curious arrangement.

On YouTube: **3Blue1Brown descibes the Dandelin Spheres**

Wikipedia also has a good description: **the Dandelin Spheres**

See how sliced cones create conic sections with these colorful foam versions:

From Ammazon: **BUY NOW: Conic Sections**

Dandelin Spheres: Slicing a cone with a plane can produce an ellipse, and two spheres encapsulated by the same cone will always have the small sphere touching one focus and the large sphere contacting the plane at the other focus. This beautiful acrylic model shows this geometry for one choice of cone width and dissecting plane angle- but it always true. This geometric construction is named for its inventor, French mathematician Germinal Pierre Dandelin back in 1822, and with it he proved theorems concerning properties of ellipses and other conic sections- mathematical entities that play a roll in much physics- including the orbits of planets. Fun math that I wish someone would have showed me back in high school!