## Introduction: Need More Jack-o'-lanterns? Maybe Archimedes Will Help Us?

The Great Archimedes is ready to help us.

这很棒！据传说说，阿基米德创造了惊人的结构，以防守锡拉丘兹的家乡与罗马帝国的战斗。

The ancient Greek scientist Archimedes discovered 13 semi-regular polyhedra.

We will use three polyhedrons as a basis for creating Jack's Halloween lanterns:

**——截断数据集;**

**- truncated octahedron;**

**- cuboctahedron。**

### 用品：

- 3 sheets of paper;

- scissors;

——胶;

- pen and ruler.

## Step 1: Preparation.

The polyhedrons' size is chosen to fit into a sphere with a diameter of 100 mm.

This will allow us to create a collection of models of comparable sizes.

最重要的是，多面体形状网将能够适合一个A4张。

Three more will be added to our collection to the two polyhedrons already created in the previous article.

To build, we need approximately**30 minutes**of your time to create each polyhedron.

## Step 2: Assembling the Cuboctahedron

For convenience, the fold lines can be drawn with a ballpoint pen.

This will make the fold lines more precise.

Cut the shape net with ordinary scissors along the contour.

Bend all the elements of the net along the fold lines.

The fold should be done inward.

胶合网的过程很简单。

The petals are marked with numbers indicating the gluing sequence.

## Step 3: Assembling the Truncated Cube.

For convenience, the fold lines can be drawn with a ballpoint pen.

This will make the fold lines more precise.

Cut the shape net with ordinary scissors along the contour.

Bend all the elements of the net along the fold lines.

The fold should be done inward.

胶合网的过程很简单。

The petals are marked with numbers indicating the gluing sequence.

## Step 4: Assembling the Truncated Octahedron

For convenience, the fold lines can be drawn with a ballpoint pen.

This will make the fold lines more precise.

Cut the shape net with ordinary scissors along the contour.

Bend all the elements of the net along the fold lines.

The fold should be done inward.

胶合网的过程很简单。

The petals are marked with numbers indicating the gluing sequence.

## Step 5: 13 Archimedean Polyhedra

The names of polyhedra are no longer as simple as those of the five Platonic solids.

Most likely, you will read the names for a long time.

But if you don't remember, then you can understand what we are talking about if we represent these polyhedra as a combination of polygons:

1. Truncated tetrahedron = 4 hexagons + 6 triangles

2. Truncated octahedron = 8 hexagons + 6 squares

3. Truncated icosahedron = 12 pentagons + 20 hexagons

4. Truncated cube = 6 octagons + 8 triangles

5.截短十二锭= 12个脱蚁饼+ 20三角形

6. Cuboctahedron = 8 triangles + 6 squares

7.Icosidodecahedron = 12 pentagons + 20 triangles

8.Great rhombicuboctahedron = 6 octagons + 8 hexagons + 12 squares

9. Truncated icosidodecahedron = 12 pentagons + 20 hexagons + 30 squares

10.Small rhombicuboctahedron = 18 squares + 8 triangles

11.Small rhombicosidodecahedron = 12 pentagons + 20 triangles + 30 squares

12. Snub cube = 6 squares + 32 triangles

13. Snub dodecahedron = 12 pentagons + 80 triangles

Maybe there is a ready-made kit so that I don't cut out the parts, but just glue it together?

Yes, there is such a set. Called Magic Edges 18, 19, 21, 27, 29, 32. It can be found on Amazon:

https://www.amazon.com/dp/B07FF3R1GB

Why exactly 13 polyhedra?

Some people have bad associations with this ominous number 13.

We have no answer to this question. Most likely, there were circumstances that exactly 13 polyhedra with such properties were discovered. Have there been any attempts to discover the 14th polyhedron with the same properties? Yes, many scientists have been looking for an answer. Some believe that the 14th polyhedron has been found. This is a Small rhombicuboctahedron with a rotated upper dome.

But if you remain a fan of the magic of the number 13, then you are unlikely to agree to such innovations.

为了增加效果，我们在万圣节前13天发布这篇文章！

在上一篇文章出版后13天。

Happy Halloween!

## 3 Discussions

3 months ago

Very nice idea. A mixture of mathematics and tradition. Thank You.

Reply 3 months ago

Thanks for your feedback. Thanks for the support!

Reply 3 months ago

You are welcome.