### 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** **

### 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!

### Mirror Anamorphic Lenticular Cup & Saucer

Direct message Luycho on Instagram about this design. Other designs can be seen here:

From Luycho:** **

Luycho | A New world on Mirrors

Click here for more Mirror Anamorphic

Mirror Anamorphic Lenticular Cup & Saucer: a toucan sits in a nest of flowers, revealed only when the cylindrical mirrored cup is put in place. This beautiful design by Luycho uses both mirror anamorphic reflection and an accordion type lenticular dual image. Art meets math and physics! @luycho for details.

### 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.

### Pentominoes

Get this set here:

From Etsy: **BUY NOW Hardwood Pentominoes **

Many versions available here:

From Amazon:** BUY NOW Pentominoes **

The book by mathematician Solomon Golomb that started the polyonomo recreational math craze:

From Amazon: **BUY NOW: Polyominoes **

Pentominoes: The 12 possible arrangements of five identical squares joined edge to edge. Since 5x12=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). I love this beautiful set from artist/woodworker Ron Moore where each pentomino is made from a different kind of hard wood.

### 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.

### 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).

### Satisfying Hexagons

Get this 3D print here (your choice of colors):

From Etsy: **BUY NOW: Satifying Hexagons**

Satisfying Hexagons: this delightful kinetic art manipulation toy features 19 nested hexagons within a hexagonal frame. Embedded magnets allows one to move the central hexagon from behind creating interesting visual effects. A 3D print created by @i.am.the.lazy.engineer- indeed oddly satisfying!

### 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.

### 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.

### The Random Walker

Galton Board version available here:

From Amazon: **BUY NOW **

Galton Board

The Random Walker: second model of two Galton Boards designed and produced by IFA.com- this version is made to demonstrate probability in investment returns of a global stock market portfolio relating to risk capacity. Slow motion reveals the erratic path of each steel ball (second half of video). The red graph shows the distribution of 592 monthly returns (mean =1%, SD=5%) representing data from 50 years of an IFA Index fund- here the random “walk” of 3000 steel balls falling through 12 levels of branching paths always produce a close match, and both distributions tend toward the famous bell curve distribution. A wonderfully 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.

### 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.

### 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).

### Hexa Sphericon

Sphericon and Hexa-sphericon: beautiful works of art in metal- available here!

From the Matter Collection:** BUY NOW The Sphericon (Hex and Regular) **

3D printed as well as handmade sphericons and similar shapes avaiable here:

From Etsy:** BUY NOW: Sphericons **

Hexa-Sphericon: Sphericons are unique solids that roll in such a way that every point on their surface comes in contact with the plane- following the path shown here with white paper. Solids from the sphericon family all have one side and two edges. Each sphericon is based on a regular polygon, with the basic sphericon derived from a square, and here- a more interesting case with more complex rolling motion- from a hexagon.

### Mirror Anamorphic Lenticular Cup & Saucer

Message Luycho on Instagram about this design. Other designs can be seen here:

From Luycho:** **

Luycho | A New world on Mirrors

Somtimes available here:

From Amazon**: BUY NOW: Anamorphic Cups & Saucers**

Click here for more Mirror Anamorphic

Mirror Anamorphic Lenticular Cup & Saucer: a flamingo sits in a nest of flowers, revealed when the cylindrical mirrored cup is put in place. This beautiful design by Luycho uses both mirror anamorphic reflection and an accordion type lenticular dual image where turning the plate 180 degrees trades images- using my new photography turntable to nice effect. Art meets math and physics!

### 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

### 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).