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

[kperry at blinksoft.com]

I love these descriptions good job.   I have magnets that make  a
tetrahedron Pyramid and one that makes  a regular polyhedron that has a
hexicgon as the shape on all sides.  I have not looked up what that is
called but it is shaped like a ball.  I will have to go up and build it and
see how many sides it has.

 

 

From: Tactile-Talk <tactile-talk-bounces at nfbnet.org> On Behalf Of John
Miller via Tactile-Talk
Sent: Thursday, October 17, 2024 7:21 PM
To: tactile-talk at nfbnet.org
Cc: John Miller <johnmillerphd at hotmail.com>
Subject: [Tactile-Talk] Looking at 3D shape such as a polyhedron

 

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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