A 78- or 79-Sided Polygon/Polyhedron

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 
or explanation please write back.]

   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, 
start with a line of length one centimeter (or any other 
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.

   I am not exactly sure that this answers your 
questions.  I hope that it is a little bit helpful.  If you 
need more information, please write back to us.  Also I 
am not very familiar with a variety of geometry programs.  
The only one that I know of is called Geometer's 
Sketchpad, and it is excellent.

        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, 
and if you'd like more information about it, write back to us 
(i.e. if you've got access to some really spiffy computer).

-Ken "Dr." Math