Tuesday, October 7, 2025

Exploring N-Gon Prisms: From TurtleStitch Code To Paper Templates


N-Gon Prisms
Starting at the bottom row and circling counterclockwise are eight prisms, 
ranging from ten-sided to three-sided.

In this blog post, I continue my quest to create prisms using paper templates. Please see my previous post to learn how my journey began: Exploring the Five-Color Torus: A Mathematical Approach.

Inspired by the elegant modular structure of that model, I became curious about designing paper versions of n-gon prisms. What began as a simple experiment quickly evolved into a fascinating mathematical puzzle.

Through trial and error, I discovered something remarkable: odd-sided prisms behave fundamentally differently from even-sided prisms.

The Odd-Sided Pattern

For odd-sided prisms (triangles, pentagons, heptagons, etc.) with an edge length of a:

  • Each face consists of two equilateral triangles connected by a rectangle.
  • The total number of equilateral triangles needed is 2n (where n is the number of sides).
  • All triangles have sides equal to the chosen edge length a.
  • The height of the prism equals the height of the equilateral triangle.

The Even-Sided Pattern

For even-sided prisms (squares, hexagons, octagons, etc.) with an edge length of a:

  • Each face is made of two isosceles right triangles connected by a rectangle.
  • These triangular pieces form squares when paired together on each side.
  • The total number of isosceles right triangles needed is 2n.
  • The height of the prism equals the side length a.

Shared Properties of Even and Odd Prisms

  • Each paper unit consists of a rectangle flanked by two triangles—one at the top and one at the bottom.
  • The rectangle is folded along its diagonal to form the three-dimensional structure of the prism.
  • The prism exists in two chiral forms, depending on whether the rectangle’s diagonal slants clockwise or counterclockwise.
  • The rectangle’s diagonal represents the longest internal diagonal of the prism.
  • The two sides of the rectangle form another diagonal, connecting adjacent sides of the prism (traversing from the top of one face to the bottom of the next, depending on the prism’s chirality).

Formulas Used

For a regular n-gon prism with side length a and height h:

Height Calculation
Odd n-gon: h = a × √3 ⁄ 2 (height of an equilateral triangle)
Even n-gon: h = a (two isosceles right triangles form a square face)

Circumradius of the Base Polygon
R = a ⁄ (2 × sin(180 ⁄ n))

Maximum Base Chord
Dmax = 2R × sin((n ⁄ 2) × 180 ⁄ n)

Longest Internal Diagonal of the Prism
Dspace = √(Dmax2 + h2)

Rectangle Dimensions
Side 1: a (polygon edge)
Side 2: √(Dspace2 − a2)

Reference Table: Paper Template Dimensions

Here is a PDF containing calculated dimensions for n-gon prisms with edge length a = 2 units:
Download PDF

Bringing It to Life: TurtleStitch and Silhouette

Using the dimensions from the table, I coded the prism nets in TurtleStitch, a visual programming environment ideal for geometric design. TurtleStitch made it possible to automate the creation of these intricate folding patterns.

I exported the designs as DXF files and opened them in Silhouette Studio software.
(Note: TurtleStitch’s DXF export doesn’t preserve scale, so resizing was necessary.)

Using the Silhouette cutting machine, I:

  • Resized the nets to the desired dimensions.
  • Added decorative windows to each panel.
  • Cut the required number of nets for each n-sided prism.

The final results were stunning—geometric forms that twist and turn, catching the light through their windowed panels.

Taking It One Step Further with TurtleStitch

Since I now understood the necessary calculations, I generalized my TurtleStitch code to eliminate the need to consult the measurement table manually.

Here is that TurtleStitch project:
Net for N-Gon Prism

With this program, creating a prism net is simple: enter the number of sides (n) and the side length (a), and the program automatically generates a net ready for cutting with Silhouette software.

After exporting as a DXF file, I opened it in Silhouette Studio, used the one-inch block for reference, and resized the entire design accordingly. Then, I used the Silhouette cutting machine to cut the required number of nets for each n-sided prism.

Assembly Instructions

Step 1: Connect the Faces

You will have n identical face pieces, each consisting of: Triangle – Rectangle – Triangle.
Fold all rectangle diagonals first—this step is critical! 
Make sure they are all facing the same directions and their orientation is the same.

Apply glue to the side of the rectangles as shown above

  1. Take two pieces and align them so their rectangle sides meet.
  2. Glue these sides together, carefully matching the corners.
  3. Continue adding and gluing each new piece in the same manner until all n faces are connected.

Step 2: Close the Tube

Join the first and last pieces by gluing their remaining rectangle sides together.

Step 3: Attach the Triangles

Valley-fold the triangles toward the rectangle’s diagonal.
Tape the adjacent triangle sides together.
Repeat until all triangles are joined, forming the complete prism


A ten-sided prism
It is amazing to see all of the diagonals of the prism converge at the center to make a hole in the middle of the prism.  I made each side a different color so that you can appreciate the structures that are formed. 
A multi-colored circle is formed with the intersection of the diagonals.  If you look carefully at the photo, you can see the longest internal diagonal.  A light pink diagonal goes from the top center of the photo to the bottom right below the light blue diagonal at the top of the prism. Try to see if you can find all ten of the internal diagonals.

The finished prisms are captivating—geometric forms that twist and turn. Each one feels like a small architectural sculpture, bridging art, craft, and mathematics.

Many thanks to: 

  • Alba Málaga Sabogal, Alix Kremer, Djatil Krichenane, Samuel Lelièvre, Richard Schwartz and Ulrich Breh for inspiring this exploration with their beautiful torus structures 
  • Saul Stahl for his foundational work on map coloring 
  • ICERM (Institute for Computational and Experimental Research in Mathematics) for hosting the Illustrating Mathematics program 
  • Claude (Anthropic) for patient assistance with the mathematical calculations and HTML development.  I am still working on the HTML development.  I will try to include it in another post.
  • The TurtleStitch community, especially Cynthia Solomon, who encouraged me to continue my quest of prisms and using TurtleStitch as a tool for mathematical making
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