### 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.

### 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.

### 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.

### Spherical Dice

A must for any die/dice collectors:

From Amazon:** BUY NOW Spherical Dice **

Click this link for other amazing dice featured on @physicsfun

Spherical Dice: these fair six "sided" dice are hollow inside with a ball that weights each sphere such that one of the six values is always on top. When these dice are rolled (literally!) the internal weight lands in one of six cavities inside creating a low center of mass which aligns one of the numbers to the top. Another low center of mass toy!

### Hyperboloid Spinner

Available here:

From Amazon: **BUY NOW **

Hyperboliod Spinner: The HypnoGizmo

From eBay: **BUY NOW **

Hyperboloid Spinner: HypnoGizmo

Hyperboloid Spinner: the HypnoGizmo toy consists of a set of slanted straight nylon lines arranged to form the outline of a hyperboliod- the quadratic surface related to the revolution of hyperbola around its axis of symmetry. As the device rotates the beads slide along in succession on one of the straight paths leading to the complex visual display. So much fun math in this toy!

### 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.

### Trammel of Archimedes

Get similar devices here:

From Etsy:** BUY NOW **

Trammel of Archimedes

From eBay:** BUY NOW **

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.

### 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).

### Ultimate Solid of Constant Width

Available in three metals and two finishes.

From The Matter Collection:

**Order NOW: Ultimate Solid of Constant Width- Brass**

**Order NOW: Ultimate Solid of Constant Width- Steel**

**Order NOW: Ultimate Solid of Constant Width- Copper**

Ultimate Solid of Constant Width: Reuleaux tetrahedrons with specially calculated curved edges become volumes of constant width- possibly the minimum volume that can possess this property. The shape featured here is a new discovery of a solid of constant width that has perfect tetrahedral symmetry. 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 remains parallel to the desktop as the orbiforms roll in between. Currently available on Kickstarter from my friends at the Matter Collection : Reuleaux tetrahedrons with specially calculated curved edges become volumes of constant width- possibly the minimum volume that can possess this property. The shape featured here is a new discovery of a solid of constant width that has perfect tetrahedral symmetry. 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 remains parallel to the desktop as the orbiforms roll in between. Currently available from my friends at the Matter Collection

### Coins of Constant Width

These triangle coins are often available on eBay.

Fron eBay: **Search NOW: Bermuda Triangle Coin**

The silver proof versions can be expansive, but sometimes the circulated coins (like those in the video) are available at a lower price.

Click to see more: **shapes of constant width**

Coins of Constant Width: the only coins produced in the shape of the Reuleaux triangle, issued in 1997 and 1998 by Bermuda. The special convex shape of the Reuleaux triangle will roll, because like a circle they have the same diameter from one side to the other, no matter their orientation. To demonstrate this property note here how two straightedge rulers remain parallel as the coins rotate between them, just as one would expect circles to behave! Bermuda triangle (ha!) coins were only produced for two years and featured Elizabeth II on the front and shipwrecks on the back- this one depicts the wreck of the Sea Venture.

### 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.