Download polyhedron.exe and polyhelp.txt to the same directory prior to running.
This application may have some use for students of solid geometry.
It is a simulation of a conducting sphere with a finite number of electrons placed on its surface. When the application is started, the user is asked for the number of electrons (points) to be used.
The program places the points randomly on the sphere, and all are allowed to skid around on the surface until their mutual repulsion forces are minimized.
Points are displayed as dark gray and light gray circles. Dark points are on the visible side of the sphere, light points are on the hidden side.
After waiting a while for them to settle in place, the user can click on the screen to connect nearest neighbors with straight lines, with the object of having the lines describe a polygon of n vertices. There are only a limited number of choices which will yield a regular polygon. With 4 points, a tetragon is formed. Six points yields an octagon. Eight points does not yield a cube, as one might expect. See what you get. Twelve points gives an isocohedron (20 faces). Twenty points should yield a dodecahedron, but doesn't. Try 24 and 32 points. 24 points yields a figure having two possible symmetries. Which one results is up to chance. The polygon can be turned about using the six buttons (up, down, left, right, clockwise, counterclockwise).
There are two possibilities for a regular dodecahedron. One has square faces, the other has pentagonal faces. This program will not draw either. Do you know why? (a question for the student). This program does allow a special way to draw a pentagonal dodecahedron. Select 32 points. Allow to settle, and click the screen two or three times. Click on the number displayed just below the "Continue" button.
For most point selections, a regular polygon cannot be obtained, and the first click of the mouse will not yield a solid figure, as nearest neighbors are usually not equally near. Clicking the screen extends the search for near neighbors, the search distance increasing for every click. Clicking enough times will end up connecting every point to all other points, and the ability to visualize a solid surface is lost. However, clicking the "continue" button allows further settling, and another chance to start over with the connecting line operation. Clicking the "deal" button allows a new start without exiting the program. Settling down to the final solution can take some time. Be patient.
The mathematical simulation assumes an inverse-square law repulsion exists between every point and every other point. After the net forces are calculated for each point, the points are allowed to move a distance proportional to their respective forces, in the force direction. There is of course an outward component to the forces, which causes the points to leave the surface of the sphere. However, a mathematical operation is performed each iteration, which pulls the points radially inward back to the surface. The "gain" applied to the iterative process has been made inversely dependent on the number of points selected, otherwise the process would become unstable for large number of points.
A figure can be saved by pressing Alt-PrntScrn to copy the window to the clipboard, from where it can be pasted into Microsoft Paint or Word for further fooling around with. --------- Additional Features:
>> Clicking on the number displayed below the "Continue" button hides all points with 5 nearest neighbors. If you start with 32 points, this eliminates 12 of them from the nearest neighbor line drawing, leaving 20 points. Lo and behold! You see a pentagonal dodecahedron.
>> Clicking the panel showing the number of points decreases the number of points by one, without a re-deal. This is handy for quick examination of a range of points.
This program was written and contributed by Bob McClure. <Bobhelenmcclure at aol com>