[Tactile-Talk] Looking at 3D shape such as a polyhedron

[John Miller]

Hello,

What follows describes a 3-dimensional object.

I find it helpful to explore the object and confirm my understanding of the object by feeling the object.

In some cases using a 3-D printer or looking at a 3-D model can be invaluable.

When I was eleven my parents took my three older brothers and me on a road trip.

We lived in Omaha Nebraska. After spending a few days in Colorado my parents drove us to the Black Hills area in South Dakota.

There we took a tour of Wind Cave in Wind Cave National Park.

After the tour my parents let my brothers and me each choose a souvenir from a tourist shop that had some interesting rocks.

I chose a translucent white rock that had been carved into some kind of polyhedron.

As a child I liked the shape's symmetry and complexity.

Recently I found the name of the shape that the rock had been made a copy of.

>From wikipedia the shape is a cuboctahedron. That is cube without the e followed by octahedron.

Perhaps you can visit a rock shop and purchase one of these for yourself.

The shape is a semiregular polyhedron because it has faces that are two shapes.

It has fourteen faces in all, six squares and eight equilateral triangles.

Recall that Leonhard Euler was a mathematician who had low vision for much of his life and was totally blind for the last seventeen years of his life. Leonhard Euler was born in 1707 and died in 1783. He published nearly half of his works while totally blind. Euler has a formula for all convex polyhedrons relating the number of vertices, edges, and faces.

According to Euler's formula, the sum of the number of faces and vertices for a convex polyhedron are two more than the number of edges.

This particular polyhedron has  fourteen faces and twelve vertices and twenty-four edges.

The sum of faces and vertices is twenty-six which is two more than the number of edges twenty-four as expected.

It is interesting to check two common polyhedrons with Euler's rule.

The tetrahedron is a pyramid with triangular base.

It is a regular polyhedron because all of its faces are the same shape which is an equilateral triangle.



Each vertex touches three faces.

It has 4 triangle faces as its name suggests, four vertices, and six edges.

The sum of faces and vertices is eight which is two more than the number of edges six, as expected.

The cube is another familiar polyhedron.

It is a regular polyhedron because each face is the same shape which is a square.



Each vertex touches three square faces.

It has six square faces, eight vertices, and twelve edges.

The number of faces and vertices is fourteen which is two more than the number of faces twelve, as expected.

It is a challenge to describe the cuboctahedron.

The eight equilateral faces are divided into four groups of two.

Each group of two forms an hourglass shape with the point of one triangle touching the point of the other.

Each vertex touches two squares and two triangles.

Each edge is shared between a square and a triangle.

Suppose you orient the shape so that it is resting on a table with a square face down and one of the square vertices of the base oriented towards you.

In this configuration the shape has another square face on top of the shape that is parallel to the table.

It has four square faces in the vertical plane that are each standing on a vertex as would a diamond shape.

The four vertical diamond faces are parallel to the front edge of the table, the left edge of the table, the back edge of the table, and the right edge of the table.

Each edge of the square base shares an edge with an equilateral triangle.

You can trace from square base across an edge to an equilateral triangle going up to a point, from that vertex up to a second equilateral triangle with its vertex at the bottom, and then across an edge to the top square in the horizontal plane.

It is also possible to place the shape so that a triangle face is the base resting on the table.

In this configuration orient one of the triangular base vertices towards you.

The top face of the shape is now also an equilateral triangle but with a vertex pointing away from you.

The top triangle face is horizontal.

The triangular base shares edges with three square faces that rise up and out at an angle.

At each of the three vertices of the triangular base is a triangle face with a vertex pointing down and a horizontal edge at the top.

The tops of these three squares and three triangles form a hexagon around the shape half way from its base to its top.

>From the top horizontal triangle are also three squares each sharing a edge with the triangle.

The top horizontal triangle also has a triangle at each of its vertices with a vertex of that triangle pointing up and an edge of that triangle sharing an edge with the previously described hexagon. Also the three squares that share an edge with the horizontal top triangle each also share an edge with the hexagon.

I have enjoyed creating some of these shapes side by side in a 2-D plane and generating the accompanying PNG image containing the shapes that share edges.

If someone would like these PNG files, please contact me off-list and I would be happy to share.

very best,

John








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