You may be familiar with shapes known as the Platonic solids: the tetrahedron (4-sided pyramid), the cube, the octohedron, the dodecahedron, and the icosahedron. Some of these shapes can be "stellated" - that is, their sides can be extended until they intersect with other extended sides. The Great Dodecahedron is a stellation of a dodecahedron. If you're interested in more details of this shape, Wolfram MathWorld is a phenomenal resource: Great Dodecahedron at Wolfram MathWorld.
Here's the Great Dodecahedron that you can make with the instructions below.
- 30 standard business cards
- Scotch tape
One interesting fact about a Great Dodecahedron is that each of the triangles that comprise it are related to the Golden Triangle. The Golden Triangle is an isosceles triangle in which the ratio of the hypotenuse to the base is equal to phi (Φ, the Golden Ratio). Phi is an irrational constant like pi, and is frequently found in geometry and nature. It's equal to (1+sqrt(5))/2, which is approximately 1.618.
The Golden Triangle can be divided such that a smaller, proportional Golden Triangle remains. The other piece of the triangle is the same as the triangles you see in the Great Dodecahedron. If you inscribed a pentagram inside a pentagon, you would see the same triangles between the arms of the pentagon. These triangles have angles of 108°, 36°, and 36°, and lengths of ratio 1, 1, and Φ.
Our first task is to create these triangles by using standard business cards. We'd like for each business card to contain two triangles, joined along the hypotenuse, so the card can be folded along the hypotenuse. This means a 60-triangle shape like the Great Dodecahedron will require 30 business cards. We want the longest hypotenuse possible, so we'd like for the hypotenuse to go from one corner of the card to somewhere near the opposite corner of the card. We are constrained by the height of the card, so we'll use that value, along with a litte trigonometry, and the fact that a 108°-36°-36° triangle has sides of Φ, 1, 1 to figure out if we need to trim the card.
We need to find the length of one of the triangle's legs, and the length of the extra overlap that will give us a rectangle with 90° corners. We'll keep this overlapping portion, because it will be used as a tab that will be used to connect this piece to other pieces in the Great Dodecahedron. Let's find the length of that overlap first:
Now, let's use the length of the overlap and the height of the business card to find the length of the isosceles triange's leg.
We now know what size our business card needs to be. For a standard US business card with a height of 2", x=2.10 and w=0.65 (approximately), giving x+w=2.75. So, cut off the extra amount from each business card (for a standard US business card that is 3.5" wide, cut 0.75" off the card).
Next, you'll need to make three folds on each card:
- A fold from the bottom, left corner to the top, right corner. Make the fold so the text of the card, which we'll want to hide from the finished
product, is on the inside of the fold. Crease the fold, and keep the card folded for the next step. (You could also fold from the top-left to the bottom-right. It doesn't matter,
as long as you fold from the same corners on all 30 cards.)
- Notice the parts of card that overlap. Fold these pieces back, so the line of the fold lines up with an edge of the card. Crease the fold. When you let go of the card,
you will have a shape that looks like this:
- Do this to all 30 cards.
Once all of the pieces are folded, it's a matter of putting them together.
- Start with five pieces.
- Take two of the pieces. Place an edge of one card into the backward crease of the overlap of the second card. Make sure the points of both cards are touching.
- Get a piece of tape, and tape the overlap of the second card to the inside body of the first card. Inside the shape, the edge of the overlap will not meet up with the long crease of the first card. Make sure to place your tape inside the shape - that is, on the sides of the cards that have text on them.
- Add a third, fouth, and fifth card to the assembly.
- When you get to the fifth card, tape the overlap from the first card to the body of the fifth card. Your assembly should look like this:
- Take five more pieces.
- These five pieces will be connected to two neighboring pieces in the assembly you just created.
- Let's start with a piece that we'll call A. We're going to insert it between Piece 1 and Piece 2 of the assembly.
- Line up the bare edge of Piece 1 with the overlap crease of Piece A. Tape Piece A's overlap to the inside of Piece 1's body.
- Line up Piece A's bare edge with Piece 2's overlap crease. Take Piece 2's overlap crease to the inside of Piece A's body.
- Do this four more times, and you'll end up with this assembly:
- Don't worry that the untaped outer edges that seem to be floating in space. When you complete the shape, the equal force on all of the edges will tighten up the shape.
Keep following this pattern of matching bare edges to overlap creases, and taping the overlaps to the inside of the neighboring pieces. You can't go wrong. The last piece will be the hardest one to connect. You won't be able to apply tape to the inside of the shape, because once you add the last piece, the shape will be closed. You could try using glue, or you could (shudder) put some tape on the outside of the shape for that last piece.
When you're done, you'll have an amazing shape that's fun to display and play with.
- The end points of each pentagram become the middle points for a neighboring pentagram. This makes the shape seem like an optical illusion.
- If you were to close off each of the triangles created by three edges, you'd have an icosahedron.
- When looking directly at the center of one of the pentagrams, the background between the points of the star are all aligned on the same plane.
- Each of the 90 triangles seen in this shape can be derived from the Great Triangle - they are isosceles triangles with edges phi, phi, and phi+1.