# Find remaining vertices of cuboctahedron.

• Eric Belcastro
In summary, the conversation discusses finding the remaining vertices of a regular cuboctahedron, given the locations of three vertices. The conversation also mentions using Maple's functions for defining and exploring archimedean solids, and using quaternion rotation matrices to rotate about known vertices and generate new vertices. There is also a correction made to one of the coordinates given in the original post.
Eric Belcastro
*note - this is not a homework problem.

I have the locations of three vertices of a regular cuboctahedron with edges of unit length (all vertices are length 1 from the center).

They are (1,0,0), (1/2, sqrt(3)/2, 0), (1/2, sqrt(3)/6, sqrt(2)/2)
or in spherical coordinates (1, 0, pi/2), (1, pi/3, pi/2), (1, pi/6, arctan(sqrt(2)/2)) respectively.

Now I am trying to find the remaining vertices of the cuboctahedron.

I believe I can find some of them easily, but I would like to be 100% certain of all of their accuracy. So I went to use maple's functions for defining and exploring archimedean solids, and to define an archimedean solid, you are only allowed to define one point, the center, and the radius, you can't specify any other vertices or angles.

It is really a simple problem, but I don't quite trust my intuition. I would assume I could just add angles where appropriate that correspond to the symmetries of a cuboctahedron and then convert back to cartesian coordinates. Any ideas of how to do this simply, or in maple?
If it is simple to do in maple, that would be nice, because there are many problems like this that pop up all the time when I am exploring something.

I figured out how to find them, it was fairly easy. I just used a quaternion rotation matrix and rotated about the known vertices according to the known rotation symmetry of a cuboctahedron and did this a few times, generating new vertices, and all was well. I also wrote one of the coordinates wrong in the original post. The third vertex was (1/2,sqrt(3)/6,sqrt(6)/3.

## What is a cuboctahedron?

A cuboctahedron is a three-dimensional geometric shape that is made up of eight triangular faces and six square faces. It is also known as an octahedron cube or a rhombic dodecahedron.

## How do you find the remaining vertices of a cuboctahedron?

To find the remaining vertices of a cuboctahedron, you can use the following formula: n = 14 + 6√2, where n represents the number of vertices. This will give you the total number of vertices, and then you can determine the remaining vertices by subtracting the known vertices from the total.

## Can a cuboctahedron be created using paper?

Yes, a cuboctahedron can be created using paper by following specific folding and cutting instructions. There are also templates available online that can be printed and assembled to create a paper cuboctahedron.

## What are the properties of a cuboctahedron?

The properties of a cuboctahedron include having 12 faces, 24 edges, and 14 vertices. It is a convex polyhedron, meaning all of its faces are flat, and it has an equal number of faces meeting at each vertex. It is also a dual of the rhombic dodecahedron.

## What is the significance of a cuboctahedron in science and mathematics?

The cuboctahedron has various applications in science and mathematics, including in crystallography, where it is a fundamental shape for certain crystal structures. It is also used in computer graphics and simulations, as well as in architectural designs. In mathematics, it is considered a semi-regular polyhedron and has interesting geometrical properties that make it a topic of study and research.

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