### The Holoscope : Icosahedron

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

The Holoscope Icosahedron: the intricate beauty of multiple internal reflections from 20 triangular mirrors in the shape of this famous platonic solid. The interior is viewed from one corner and illuminated by light entering from glass spheres placed at all of the other 11 vertices. A type of kaleidoscope based on mirrored polyhedra by artist Gary Allison, (swipe to see the dodecahedron and cube) and future posts will include tetrahedron and octahedron 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.

### Sphere Sticks Geometric Puzzle

Get this affordable and amazing puzzle here:

From Etsy: **BUY NOW: Sphere Sticks**

Sphere Sticks Puzzle: 30 identical wood pieces, each with two notches as shown, can create 12 interlocking pentagons in a perfect symmetry- look carefully and you can see that each rod is in an identical configuration with the 4 others that connect with it. Precision cut notches on the rods allow them to interlock with elastic tension such that vector sum of the 4 forces sum to zero in this tensegrity type equilibrium. The dodecahedron, with its 30 edges and 12 sides, is the basis of this puzzle sculpture.

### High Voltage Fractal in Wood

Amazing creations made here:

From Etsy store EngravedGrain:** BUY NOW High Voltage Fractal **

High Voltage Fractal in Wood: a Lichtenberg fractal created by a high voltage electrical current flow across a piece of wood. Since wood is an insulator a light coating of conducting water (for instance a solution of baking soda or salt) is first applied to the surface. Metal electrodes are then attached at each end of the wood piece and a dangerous source of high voltage is applied (such as a microwave oven transformer or neon light transformer).

### Ambiguous Object

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 **

Similar objects available here- from Etsy:** BUY NOW**: ** Ambiguous Objects**

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.

### 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.

### 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.

### Platonic Solids Dice Set

Get this nice set of regular polyhedra dice here:

From Etsy: **BUY NOW: Polyhedra Dice Set**

Less expensive sets of standard plastic dice here:

From Amazon: **BUY NOW: Polyhedra Dice Set**

Platonic Solids Dice Set: The five famous convex regular polyhedra in the form of fair dice. This set is cast in metal with clean edges and makes for a great way to own these symmetrical objects that have fascinated thinkers ever since the ancient Greeks wrote about them circa 360 BC. Only these five forms meet these criteria (in 3D space) for each face: must be equal in size, be equal in number of sides, each side of equal length, identical in angle were any two sides meet, and have the same number of sides meet at each vertex point of the solid. 2000 years later the famous mathematician Euler determined that for these 5 shapes V-E+F=2, the number of corners (vertices), minus the number of edges, plus the number of faces, will always equal 2.

### Spherical Dice

A must for any die/dice collectors:

From Amazon:** BUY NOW Spherical Dice **

From eBay:** 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!

### 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.

### Anamorphic Harry Potter Puzzle

No longer in production but available on eBay:

From eBay: **BUY NOW: Harry Potter On Reflection Puzzle**

Mirror Anamorphic Harry Potter: the conical mirror of the silver container reflects and reveals the distorted scenes in this 200 piece jigsaw puzzle of the famous wizard and school. The shape of the mirror allows for a mathematical operation, a type of affine transformation, to map the distorted image of the puzzle to the restored image reflected by the mirror. Real magic!

### Holoscopes

Order a holoscope from the artist's gallery here:

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

6 and 12 Faced Holoscopes: a cube and dodecahedron of mirrors. A type of kaleidoscope based on truncated Platonic solids by artist Gary Allison @holoscope2000. These holoscopes are in the form of two famous Platonic solids- each with all the corners cut off to allow light to enter through triangular holes. Through the symmetry of the reflections, these triangles appear as an infinite array of tetrahedron shapes in the dodecahedron, and as beautiful repeating stellated octahedrons in the cube. Each holoscope has stained glass on the exterior and front surface mirrors on the inside which create the amazing and seemingly impossible spaces within.

### 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 2020.

### Orbiforms

Orbiforms available here:

From Etsy: **BUY NOW Orbiforms**

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. Featured items from @altdynamic** **

### Curves of Constant Width

Get a set here:

From Maths Gear:** BUY NOW Curves of Constant Width (set of 4) **

Click here for 3D **Solids of Constant Width**

Curves of Constant Width: regardless of the precise shape, any curve of constant width has a perimeter equal to pi times its width! These convex shapes will roll because like a circle they have the same diameter from one side to the other no matter their orientation. Here are two famous examples: the Reuleaux triangle (found in rotary engines) and a Reuleaux pentagon- note how the two straightedge rulers remain parallel as the shapes rotate between them, just as one would expect circles to behave! These physical representations of the special curves seen here are produced by Maths Gear (Matt Parker and Steve Mould).

### Hyperboloid Spinner

Kit available here:

From Amazon: **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!

### Squaring Circles

This sculpture available as a 3D print:

From ShapeWays: **BUY NOW: Square Circle Illusion**

See other amazing geometric illusions here: **Ambiguous Objects**

Squaring Circles: from one particular point of view these wireframe sculptures looks like a circles/squares, from another it’s a square/circle! From other viewing angles one can see that the underlying curves are four identical segments of a parabola conjoined. Further examples of how a single perspective can be misleading! Math sculptures available as a 3D print by Matt Enlow.

### 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!

### 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).