Written by Dave Martin
• 6/24/2024
• Read Time : 4 min.

Mathcad Community Challenge May 2024: Platonic Solids

The May 2024 Mathcad Community Challenge was inspired by Dungeons & Dragons and the Netflix show Stranger Things:

D&D involves spinning dice, which are regular convex polyhedrons. Common dice include the d4 (tetrahedron), d6 (cube), d8 (octahedron), d12 (dodecahedron), and d20 (icosahedron), where ‘d’ is for die and the number indicates the number of sides. Platonic solids are regular polyhedrons made up of regular polygons (the lengths of each side and interior angles are equal).

• Derive the surface area of the regular polyhedrons as a function of the radius of a circle that circumscribes a polygonal face.
• Create a calculator where the user can select a regular polyhedron by its name or number of sides, and for a given radius of the circumscribed circle of the polygonal face, the surface area and volume will be calculated.
• After the user selects a polyhedron, the number of vertices, number of edges, and Euler characteristic for the polyhedron should be displayed on the worksheet.
• Using the Chart Component, graph the surface area and volume as a function of the number of faces.

The submissions

For this challenge, four people submitted five worksheets. Let’s take a look at their responses.

Frequent contributor Tokoro was the first to submit a worksheet. He submitted a second one later in the month. Right off the bat, his worksheet takes advantage of the new advanced input controls in Mathcad Prime 10, including the slider (to change the radius of the circumscribed circle), a check box to fill the surfaces, and a list box to select one of the various polyhedrons. The second submission included more check boxes to plot the circle and the radius.

If you view Tokoro’s worksheets, be sure to look at what he has in the Draft view. He uses a Table for all the polyhedron properties, including functions for the edge length, surface area, and volume. (I didn’t know you could include functions as table entries. Neat.) There’s symbolic evaluation including substituting the circumscribed circle radius for the edge length. The main part of the worksheet plots each of the polyhedrons in their own 3D Plot and uses an XY Plot for surface area and volume as a function of the number of faces.

Alan Stevens starts with the formula for the area of a polyhedron as a function of the number of faces, number of edges per face, and radius of the circumscribed circle. He cleverly wrote a function that returns the parameters of a polyhedron based on its first letter and a Boolean comparison. A second function returns the polyhedron properties based on the first letter and radius. He then created an XY Plot that shows a column trace of the area as a function of the number of faces.

PPal’s worksheet starts off with documentation and images; that’s always nice. You can select the shape from the new List Box input control. Then a program calculates the polyhedron properties based on the selection. Those properties are reported in a matrix. PPal uses the currency symbol as a placeholder for the length dimension. Interesting; I had never seen that used before.

Patrick H defined a series of programs like a set of subroutines. The last program in that series calls the subroutines to calculate the characteristics of a polyhedron. He also used the new List Box input for selecting the polyhedron. The surface area and volume as a function of the number of faces were graphed using the Chart Component with a second Y-axis.

Then for bonus points, he wrote programs to make 3D plots of each of the Platonic solids. I recommend downloading this worksheet so you can click on each of the 3D plots and spin the polyhedrons around.

Next steps

I did not expect to see use of the advanced input controls so early. I am personally very excited about this Mathcad Prime 10 functionality. It expands the ability of users to create worksheets that serve as reusable tools for engineering design work, and help educators make worksheets to serve as teaching tools.

If you would like to check out the new advanced input controls, download a free trial of Mathcad Express. You will get full Mathcad functionality for 30 days (including advanced input controls), and then it will revert to the Express version.

As always, if you want to expand your skills with Mathcad, download the worksheets from this month and previous months. I guarantee you will learn a lot!