### 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: this post celebrates pi day with coins in the form of Reuleaux polygons- shapes for which their perimeter/diameter = π, just like the circle! The special convex shapes of the Reuleaux triangle (Bermuda 1 dollar) and Reuleaux heptagon (UK 50 pence) 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 how two straightedge rulers remain parallel as the coins rotate between them, just as one would expect circles to behave. Bermuda triangle (ha!) coins are the only coins produced in the shape of the Reuleaux triangle, issued in 1997-98 and featured Elizabeth II on the front and shipwrecks on the back.

### 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!

### Sphere and other Orbiforms

Similar items available here:

From Etsy: **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.

### 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).

### 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 recent 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

### Volumes of Constant Width

Get a set of Reuleaux solids here:

From Etsy: **BUY NOW Volumes of Constant Width **

Volumes of Constant Width: 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 top. Two of these three solids are based on the Reuleaux triangle and the other on a Reuleaux pentagon.

### 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).

### 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** **