### Equilateral Triangular Kaleidoscope

This inexpansive kaleidoscope is available here:

From increadiblescience: **BUY NOW: Moire Tube Kaleidoscope**

Click here for affordable, precision made scopes with angled mirrors: **Kaleidoscope Symmetries Explored**

See more kaleidoscopes in my collection: Kaleidoscopes

Equilateral Triangular Kaleidoscope: three mirrors arranged in a 60-60-60 degree triangle creates the appearance of a plane filled with triangles (or equivalently a honeycomb lattice)- perhaps the most common mirror configuration design, this inexpensive kaleidoscope produces an excellent example of the reflection pattern. As a bonus the exterior tube on this scope incorporates a kinetic Moirè pattern. The kaleidoscope was invented by the famous Scottish physicist Sir David Brewster (1781-1868), and has become an entire field of artistic endeavor.

### 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.

### Mathematica Eames Souvenir

Wikipedia has **great info about Mathematica** and where to see it.

The Poincare Disk is a representation of non-Euclidean gemometry and tesselations can look amazing- M.C. Escher used this math for his famous Circle Limit woodcut.

Mathematica features a large **Galton Board (click to get one)**.

Mathematica Eames Souvenir: from the 1961 opening of “Mathematica: A World of Numbers and Beyond” by the famous design team of Charles and Ray Eames this souvenir of card stock features a tessellation of the hyperbolic plane (a Poincaré Disk) with triangles on the front and on the reverse is the title of the exhibit but hidden within an anamorphic font. Amazing math art before the age of computer graphics! There are three versions of the exhibit and all can still be seen at museums in Boston, Atlanta, and Dearborn.

### Motion of Rolling Sphericons

Get these and other amazing developable rollers here:

From Etsy: **BUY NOW: Hexacon and Sphericon Rollers**

*Note: this site contains affiliate links for which I may be compensated*

Sphericons are unique solids that roll in such a way that every point on their surface comes in contact with the plane. Solids from the sphericon family all have one side and two edges. Each sphericon is based on an even sided regular polygon, with the basic sphericon derived from a square, and the next two possible shapes are the hexa-sphericon and octo-sphericon, each rolling with a more complicated path.

### Pencil Hyperboloid

Choose your color and get one here:

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Hyperboloid Pencil Holder

don't forget a set of pencils:

From Amazon: **BUY NOW **

Colored Pencil Sets

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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!

### Hyperboloid Spinner

Kit available here:

From Amazon: **BUY NOW **

Hyperboloid Spinner: HypnoGizmo

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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!

### Kostick Stars

Two versions available here:

From Art of Play:** BUY NOW: Kostick's Stars **

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Here is the detail of the connections of the 6 star from the patent document:

Learn more about John Kostick here: John and Jane Kostick handcrafted mathematical constructions

Kostick Stars: The geometric forms of these kinetic sculptures snap into place due only to the balanced tension forces between thear slightly bent bronze wires. The large star is comprised of six struts, each made of 5 wires, which are made to intersect and weave though each other with beautiful geometric regularity. Similarly the small star has 4 struts each made of 3 wires. The ends of the 6 struts of the large star become the 12 vertices of an icosahedron, and the ends of the 4 struts of the small star point to the 8 corners of a cube. Both stars can be collapsed along many of the symmetry axes. Invented and patented in 1970, these stars are still produced and sold by artist and designer John Kostick.

### Chiral Aperiodic Monotile Puzzle

Get this amazing puzzle set here:

From Etsy: **BUY NOW: Specter Tile Set by Nervous Syestem**

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Read the published paper: A chiral aperiodic monotile

Chiral Aperiodic Monotile: these puzzle pieces are all identical in shape, based on special family of forms called “specters”, and are key to a major mathematical discovery in May of 2023. Specters will tessellate a plane to infinity, similar to how squares or hexagons can tile out with no gaps. However specters are identically shaped tiles which fill the plane aperiodically- if one tries to shift a part of a specter tiling, the shifted part will not align or match up with any other part of the same tiling- all the way out to infinity! The fact that aperiodic tessellations exist at all is pretty amazing, and Sir Roger Penrose (Nobel prize in physics 2020) is also famous for discovering a pair of regularly shaped polygons in the 1970s that tile in this aperiodic way. Specters were discovered by the same team of mathematicians (Smith, Myers, Kaplan, and Goodman-Strauss) who discovered “the hat” aperiodic monotile in March 2023, but the hat needs its mirror image to complete the tiling whereas the specter does not. An amazing development found only two months after the initial major breakthrough. Credit to the @nervous.system design team for modifying the specter into an interlocking puzzle piece form- and assembling these without gaps is a fair challenge!

### Trisected Sphere Puzzle

Get one here (color of your choice):

From Etsy: **BUY NOW: Trisected Sphere Puzzle**

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Trisected Sphere Puzzle: three identical pieces interlock to form a ball in this elegant 3D printed dissection puzzle. The sphere is sliced following a helical path from one pole to the opposite pole, creating a smooth disassembly and reassembly.

See other fun geometric puzzles here: dissection puzzles

### Shashibo Puzzle Dissected

Get this cube puzzle (in a variety of colors and patterns) here:

From Amazon: **BUY NOW: Shashibo Cube**

From Art of Play**: BUY NOW : Shashibo Geometric Art**

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Shashibo Puzzle Dissected: I cut apart one of these puzzles to better understand the geometry: it is indeed a cube dissected into 12 equal irregular tetrahedra as seen here. The embedded magnets still allow construction of the cube. Of course the symmetrically placed hinges (along with strategically placed magnets) creates the real magic of the Shashibo puzzle- a device with more that 70 geometrically interesting and aesthetic configurations including a rhombic dodecahedron as well as the cube.

### Cone of Apollonius 3D print

Get this inexpensive version here:

From Etsy: **BUY NOW: Cone of Apollonius/ Conic Sections**

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Cone of Apollonius 3D print: Slicing a cone with a plane will produce the famous curves known as the conic sections. 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.

### Zulu Telephone Wire Bowl

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From Etsy: **BUY NOW: Zulu Wire Bowls**

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Geometric Weaving: this Zulu telephone wire bowl, made in the traditions of South Africa, features a brilliant spiral pattern. The bright colors come from weaving strands of colored wires- the kind used to connect phones throughout buildings. This particular pattern inspired using a rotating base to display this bowl as kinetic sculpture.

### Steinmetz Solids

Get these as a 3D printed set here:

From Etsy: **BUY NOW: Stienmetz Bi, Tri, and Quadcylinders**

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Steinmetz Solids: 3D prints of the bicylinder- the space remaining from the intersection of two cylinders at right angles, and the tricylinder- the volume from the perpendicular intersection of three cylinders. Assemblies of acrylic rods illustrate the intersections that produce these shapes named after the mathematician Charles Steinmetz, who was also a physicist and electrical engineer with over 200 patents to his name from 1895-1915. Fun facts: The bicylinder casts a circular shadows along two orientations, and a square shadow perpendicular to those. The tricylinder casts a circular shadows along three orientations, and a hexagonal shadow along another three.

### Cubic Holoscope

Order a holoscope from the artist's gallery here:

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

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Cubic Holoscope: a cube of mirrors creates a geometric universe within. A type of kaleidoscope based on truncated Platonic solids by artist Gary Allison @holoscope2000. Through the symmetry of the reflections, these triangles appear as an infinite array of repeating stellated octahedrons. Each holoscope has stained glass on the exterior and front surface mirrors on the inside which create the amazing and seemingly impossible spaces.