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### A 78- or 79-Sided Polygon/Polyhedron

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Date: 13 Mar 1995 12:50:54 -0500
From: Ray White
Subject: Help

I am interested to know if it is possible to produce a
78 or 79 sided polygon, and if so what would it look
like? Also, do you know of any good software for
creating and manipulating such things?

Thanks..
```

```
Date: 13 Mar 1995 21:57:57 -0500
From: Dr. Ethan
Subject: Re: 78 polygon

Hey Ray,
I have a few questions.  Do you want a regular
78 sided polygon, meaning that all the angles are
equivalent and all the sides have the same length or do
you just want a 78 sided polygon?

If you want the latter, then an easy way to get one
is to take a ruler and draw a line one centimeter long.
Then change the angle by one degree and draw another
line at the end of the first one.  Repeat this until you
have 77 lines.  Then connect the end of the last line
with the beginning of the first line.  That will be a
weird, lopsided but 78 sided polygon.

If you want it to be a regular polygon, it will be a
little bit trickier.

Do you know that the sum of the interior angles
of a polygon of n sides is 180(n-2)?  For instance, a
triangle has 180 degrees, a square 360 degrees, etc.
[If you have never seen this and would like justification

This means that a 78 sided polygon will have
180(76) for the sum of the interior angles.  So if all the
angles have the same length, then to find the length of
one angle we can divide this total by 78.

Now we can go back to our pencil and paper.  Again,
length that is convenient).  Then draw the next line so
that it makes an angle of whatever you calculated
[180(76)/78] each angle to be, and keep repeating this.
If you are very careful, this should soon begin to look
like a circle.  By the time you get to 77, you should
just have to add the last line and it should close itself.

questions.  I hope that it is a little bit helpful.  If you
am not very familiar with a variety of geometry programs.
The only one that I know of is called Geometer's

Hope that helps

Ethan Doctor On Call
```

```
Date: 14 Mar 1995 18:38:24 -0500
From: Ray White
Subject: Re: 78 polygon

Oops, can't believe I did that. I did not meant a two
dimensional polygon, but a three dimensional shape with
78 sides.  As a cube is a 4 sided shape.  Sorry. :)
```

```
Date: 19 Mar 1995 16:54:49 -0500
From: Dr. Ken
Subject: Re: 78 polygon

Hello there, Ray!

I think I'm going to assume that was another typo there:  did
you mean to say "as a cube has 6 sides," like the faces of a die?
It has six sides and twelve edges.  In any case, you can create a
polyhedron (the general name for a solid object with straight
edges and pieces of planes for faces) with any number of sides
greater than 4, simply by following this method:

Start out with a solid cube.  Then lop off one corner with a big
knife.  Now you've got all six of the original faces, with one
new face.  Certainly you can do this with all eight corners to
produce an object with 14 sides.

But the fun doesn't stop there.  Notice that when you made that
cut, you sliced off a corner where three edges came together,
and you created three new corners which are also junctions of
three edges.  So you could lop these off too, and for each slice
you make, you add one new face.  In no time, you'll get up to
78 sides.

Of course, this is kind of an irregular (weird) polyhedron.  But
there is no regular (where all the faces and edges are congruent)
78-hedron.  So these kinds of wacky substitutes are all you're
going to get.

I do know of some excellent software for manipulating 3-D
objects, but it only runs on some select computers (such as an
SGI or a NEXT machine).  The program is called Geomview,

-Ken "Dr." Math
```
Associated Topics:
Middle School Geometry
Middle School Polyhedra
Middle School Triangles and Other Polygons

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