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
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