## Introduction: Zip Tie Ball

This ball is made from zip ties (zap straps) and has a nice mathematical colouring.

You will need:

- 60 small zip ties (zap straps, plastic cable ties). Large zip ties are more difficult to hack for this project.
- A push pin
- Needle nose pliers
- (optional) Tester's model paint

Many different shapes can be built with this method.

## Step 1: Make a Zip Tie Triangles

Attach 3 zip ties together in a loop, pulling the heads half way onto the "zippers". Now do it 20 times.

## Step 2: Connect Triangles

This is the secret step to connect two triangles: Stuff the thin tail from one triangle into a zip tie head on another triangle.

First, you need to decide which way your triangles go; clockwise or counterclockwise. Whichever it is,

First, you need to decide which way your triangles go; clockwise or counterclockwise. Whichever it is,

**take note of which side of the triangle will be inside the Zip Tie Ball. This is important because it will give the ball a more uniform curvature.**-
Use a push pin to widen a head's hole, above the existing zip tie inserted through it, on the edge
**near the inside of the ball**. - Hold the head and zip tie firmly in place while you remove the push pin.
- Insert a zip tie tail of another triangle into the head, keeping the same orientation and direction as the original zip tie. Use the push pin to coax it through if you need to.
- When you can get hold of it, pull it through with needle nose pliers until just before the zipper part enters the head.

## Step 3: Building Patterns

Congratulations, you have learned the basic step. Now if you do that 60 times in the right places, you'll have a Zip Tie Ball!

The Zip Tie Ball has 12 circles each made of five triangles (faces of the dodecahedron). We'll start with one circle and build from there.

Notice that each pair of adjacent triangles is attached at two places, forming a little diamond between them.

Don't just build 12 of circles. Build onto the first one instead, and close circles when they have 5 triangles.

The Zip Tie Ball has 12 circles each made of five triangles (faces of the dodecahedron). We'll start with one circle and build from there.

Notice that each pair of adjacent triangles is attached at two places, forming a little diamond between them.

**Be careful that all your triangles are oriented the same way you chose in step 2!**Don't just build 12 of circles. Build onto the first one instead, and close circles when they have 5 triangles.

## Step 4: Build Onto the First Circle

Your first circle has 5 tails, so attach a triangle to each one.

**The great thing about this shape is that it is mathematically impossible to mess it up**(but human error will almost surely thwart your first effort). As long as you follow the instructions when attaching new triangles, and all your circles have exactly five triangles around them, you cannot end up with the wrong shape (I may regret saying this). That said, I'll give you a few more pointers.- Always connect everything that should be connected before adding a new triangle to the zip tie ball.
- Be very methodical. After making the first circle, extend each tail by one triangle. Then extend one of those by one triangle, and close the new circle that has 5 triangles. Now do another one, etc.
- Double check things often. Are your circles exactly 5 triangles around? Is there something else you can connect? Should you add a new triangle, or connect existing ones?

**If you want to make different shapes, make circles with different numbers of triangles around them. You will likely need a different number of zip ties if you do that.**## Step 5: Finish the Construction

After you've closed half of the circles, most of the building is finished.

Now extend each tail by a triangle (I might have broken one of my rules here), and then attach them together to form the final circle, as well as close the five circles adjacent to it.

Now extend each tail by a triangle (I might have broken one of my rules here), and then attach them together to form the final circle, as well as close the five circles adjacent to it.

**YOU ARE DONE THE CONSTRUCTION!**## Step 6: How to Colour the Zip Tie Ball With "A5"

You only need to know how to add one colour. If you follow the same instructions for all 5, they will never interfere with each other! When I say "colour a diamond", you can take that to mean, "colour part of a diamond" as long as you are consistent for every diamond.

- Pick a colour and use the same one for all these steps.
- Colour an uncoloured diamond.
- Colour the diamond opposite this diamond.
- Imagine the Zip Tie Ball is the Earth, and the coloured diamonds are the North and South Poles. The Equator passes through exactly four more diamonds (also opposite pairs).
- Colour the diamonds on the equator.

## Step 7: Colour in the "fill" and Learn About the Symmetry Group

Colour the remaining parts of the Zip Tie Ball with a neutral "fill" colour.

This colouring is very special because it shows off

The symmetry group of a shape is the set 3D rotations that take the shape from a given position, to all those that are symmetric to it. For a pencil this is a rotation around the length of the pencil through 1/6, 2/6, 3/6, 4/6, 5/6 and 6/6 of a full rotation. We call this the cyclic group on 6 elements.

For the Zip Tie Ball it is more complicated, so we abstract the essential information to describe the symmetry group in terms of five, colours which I call 1,2,3,4, and 5. If we take the colours 12345 to be those that are clockwise around one of the circles, starting at the noon position (from wherever we are viewing the shape), we can rotate the Zip Tie Ball clockwise around that circle to make it show 51234. And then again to get 45123, and again 34512, and again 23451, and again 12345. We have already discovered a subgroup, and it is the cyclic group on 5 elements!

There are other symmetries which map circles to circles, and they are all a special kind of permutation of the original 12345, called an even permutation. It's nice that our colours are numbered, because an even permutation of 12345, is one that has an even number of digits that are out of order; that is, the larger one is on the left. For example 51342 has 6 pairs of digits out of order (51,53,54,52,32,42), so it is an even permutation. The even permutations of 12345 make up exactly half of all possible permutations, which total 120. Therefore our Zip Tie Ball has exactly 60 symmetries!

Here is a nice introduction to alternating groups for math students.

Let's go back to where we rotated around a circle. After 5 steps, we came back to where we started at 12345. Since these 1/5 circular rotations that we made brought us back to the start after being applied 5 times, we call this an element (3D rotation) of order 5. We can make 1/3 circular rotations around a triangle to get an element of order 3, and 1/2 circular rotations around a diamond are of order 2. These circular rotations can be combined.

So what does a 1/3 clockwise rotation do to 12345? Now we really need to use the Zip Tie Ball. First decide which colours will be 12345. I've chosen red,blue,purple,yellow,green in the photo. If we rotate 1/3 clockwise around the triangle directly beneath it, the colours 12345 are replaced with purple, blue,yellow,red,green or 32415. The colours not touching the triangle (2 and 5) have not changed position!

Now check out the 1/2 rotations around diamonds yourself! Can you get 3D rotations of any other order by combining different cyclic (circular) rotations?

Another exercise: What happens to the colours around other circles when you rotate around a circle. For example, if I rotate around the circle 12345, through to 51234, what happened to the colours on the circle 32415?

Not satisfied? Don't forget that it bounces!

I welcome any corrections and clarifications in the comments below. I am a PhD student in mathematics, but my knowledge of symmetry groups is purely recreational.

**Congratulations, you are finished!**Now make yourself some coffee and learn about the beautiful colouring we used.This colouring is very special because it shows off

**the symmetry group of the shape**. A symmetry of a shape is a pair of distinct positions that look the same. You've probably heard of mirror symmetry, since your face has left-right mirror symmetry, but that's exactly the kind we won't think about here. We are talking about 3D rotational symmetry. For example a pencil has 6-way cyclic rotational symmetry. There are six positions of the pencil that look the same because they are rotations of each other (assuming the pencil has no markings). Each shape has exactly one symmetry group.The symmetry group of a shape is the set 3D rotations that take the shape from a given position, to all those that are symmetric to it. For a pencil this is a rotation around the length of the pencil through 1/6, 2/6, 3/6, 4/6, 5/6 and 6/6 of a full rotation. We call this the cyclic group on 6 elements.

For the Zip Tie Ball it is more complicated, so we abstract the essential information to describe the symmetry group in terms of five, colours which I call 1,2,3,4, and 5. If we take the colours 12345 to be those that are clockwise around one of the circles, starting at the noon position (from wherever we are viewing the shape), we can rotate the Zip Tie Ball clockwise around that circle to make it show 51234. And then again to get 45123, and again 34512, and again 23451, and again 12345. We have already discovered a subgroup, and it is the cyclic group on 5 elements!

There are other symmetries which map circles to circles, and they are all a special kind of permutation of the original 12345, called an even permutation. It's nice that our colours are numbered, because an even permutation of 12345, is one that has an even number of digits that are out of order; that is, the larger one is on the left. For example 51342 has 6 pairs of digits out of order (51,53,54,52,32,42), so it is an even permutation. The even permutations of 12345 make up exactly half of all possible permutations, which total 120. Therefore our Zip Tie Ball has exactly 60 symmetries!

Here is a nice introduction to alternating groups for math students.

Let's go back to where we rotated around a circle. After 5 steps, we came back to where we started at 12345. Since these 1/5 circular rotations that we made brought us back to the start after being applied 5 times, we call this an element (3D rotation) of order 5. We can make 1/3 circular rotations around a triangle to get an element of order 3, and 1/2 circular rotations around a diamond are of order 2. These circular rotations can be combined.

So what does a 1/3 clockwise rotation do to 12345? Now we really need to use the Zip Tie Ball. First decide which colours will be 12345. I've chosen red,blue,purple,yellow,green in the photo. If we rotate 1/3 clockwise around the triangle directly beneath it, the colours 12345 are replaced with purple, blue,yellow,red,green or 32415. The colours not touching the triangle (2 and 5) have not changed position!

Now check out the 1/2 rotations around diamonds yourself! Can you get 3D rotations of any other order by combining different cyclic (circular) rotations?

Another exercise: What happens to the colours around other circles when you rotate around a circle. For example, if I rotate around the circle 12345, through to 51234, what happened to the colours on the circle 32415?

Not satisfied? Don't forget that it bounces!

I welcome any corrections and clarifications in the comments below. I am a PhD student in mathematics, but my knowledge of symmetry groups is purely recreational.

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