Orbiform: A mathematically perfect solid of constant width
Orbiform: A mathematically perfect solid of constant width

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About

An Orbiform is a little-known, unintuitive geometric shape, with fascinating mathematical properties. Orbiforms were unwittingly used by polymath Leonardo da Vinci in 1514, mathematically discovered by mathematician Leonhard Euler in 1771 and placed in application by engineer Franz Reuleaux in 1876.

We are Kickstarting a 3D Orbiform derived from a triangle to spread our enjoyment of mathematics and design with math lovers, designers, creators, educators and students.

Some of the many prototypes made in Copper, Brass and Stainless Steel.
Some of the many prototypes made in Copper, Brass and Stainless Steel.

An Orbiform is an amazingly aesthetic and elegant desktop sculpture to add to any workspace. It is an amazing conversation piece, yet also meditative to roll around. Made from high quality materials and precision machined, the stylish Orbiform is a representation of where art and math meet.

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Math Talk

An Orbiform is a shape whereby its closed curves result in a constant width. A 2D Orbiform can be rotated between two parallel, fixed distance lines and remain tangent to both lines at every orientation just like a circle of the same width. Interestingly, the perimeter length of a circle and Orbiform can both be calculated with the equation pi*D=circumference.

This 2D diagram shows how an Orbiform keeps the exact same width, in all orientations.
This 2D diagram shows how an Orbiform keeps the exact same width, in all orientations.

A 3D Orbiform is an analog to the 2D case. A 3D Orbiform can be placed inside a cube in any orientation and simultaneously, tangentially touch all six faces just like a sphere of the same width. 

This diagram shows how a 3D Orbiform can simultaneously touch all six faces of a cube regardless of how it is positioned.
This diagram shows how a 3D Orbiform can simultaneously touch all six faces of a cube regardless of how it is positioned.

It is commonly thought that only a circle or sphere has constant width, but that is not the case. Leonardo da Vinci was unaware of the Orbiform's constant width property, even while he used the shape in his own work. It would take another 257 years until the constant width property of Orbiforms was realized and mathematically proven by Euler.

Leonardo da Vinci and  Leonhard Euler
Leonardo da Vinci and Leonhard Euler

The Orbiform can roll like a sphere, even while having a point, an edge and not being a smooth curve. The main difference, is the center of mass of an Orbiform is not central like a sphere, so it won't continue rolling unless a force remains on it. 

The the flat board above the Orbiforms always stays at the same height, regardless of how it is rolled, just as if three spheres were under it.
The the flat board above the Orbiforms always stays at the same height, regardless of how it is rolled, just as if three spheres were under it.

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The Orbiform also holds a unique property, as it is the smallest shape by volume to have constant width, meanwhile a sphere is the largest. Below is the derivation of an Orbiform with width of unit 1.

Integral Solid of Revolution Method Used
Integral Solid of Revolution Method Used

The above total volume (purple plus green) is .449. A Sphere, of diameter 1, would be .52, 15.8% larger than an Orbiform.

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Precision Machined

Orbiforms are available in stainless steelcopper and brass. Each metal is precision CNC (computer numerical control) machined and then hand finished in one of three surface finishes. You can choose a satinmirror polish or machine finish. The pictures below show the nine different options.

Each Orbiform is 1.4 in. (3.55 cm) in width. Stainless Steel weighs 161 g (5.67 oz.), Brass 159 g (5.96 oz.), and Copper 180 g (6.35 oz.).

Stainless Steel with Surface Finishes: Satin, Mirror Polish, Machine
Stainless Steel with Surface Finishes: Satin, Mirror Polish, Machine

 

Copper with Surface Finishes: Satin, Mirror Polish, Machine
Copper with Surface Finishes: Satin, Mirror Polish, Machine

 

Brass with Surface Finishes: Satin, Mirror Polish, Machine
Brass with Surface Finishes: Satin, Mirror Polish, Machine

Notes on Finishes: The satin finish is just like the metal finish on an Apple computer, it is not reflective and hides finger prints extremely well. The mirror polish is literally as reflective as a mirror. You can shave in it or see what is happening behind you. We didn't call it mirror polished for nothing. The machine finish looks brushed, but it is actually a precision machined surface. The machine finish is so precise and fine, that it creates a diffraction grating, which will reflect poly-chromatic light into its constituent colors, showing a rainbow.  

A macro shot of the Machine Finish where you can really see the even and precise texture.
A macro shot of the Machine Finish where you can really see the even and precise texture.

 

Mirror Polished at every angle.
Mirror Polished at every angle.

 

Satin Finish absorbs light and has a texture similar to an Apple computer.
Satin Finish absorbs light and has a texture similar to an Apple computer.

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Production Milestone Timeline

Each dot represents a milestone. We will send project updates about these milestones as they happen to keep you updated just as we have done in our past campaigns. We will not be like other Kickstarter campaigns that leave their backers in the dark and guessing as to what is happening. Every order will also have tracking. 

 

Reach Goals

Once we get funded, we will come up with some reach goals :). To help us get there, please consider sharing this project with your friends and family!

One backer wrote in asking for a display of some kind for their Orbiform. If you have an idea you would like to see as a reach goal, please write us! 

Please keep in mind, Orbiforms are very heavy so don't drop them! They are not children's toys!

The Brass Machine Finish Orbiform with some of our other favorite gear.
The Brass Machine Finish Orbiform with some of our other favorite gear.

 

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