Department of Mathematics and Systems Analysis

## Crystal Flowers in Halls of Mirrors

Sebastian Björkman, Tristan Hamel, Zsuzsanna Horvath, Kaisa Kangas, Saara Louhensalo

## HIMMELI

The piece consists of two objects. Both of them are three-dimensional “shadows” of the same four-dimensional object called the 24-cell. In mathematics, this type of a “shadow” is called a projection.

Three-dimensional polyhedra can be built using two-dimensional faces. For example, a cube consists of six square-shaped faces. Similarly, many four-dimensional objects have three-dimensional polyhedra as their building blocks. The 24-cell consists of 24 three-dimensional cells that have an octahedral shape.

If you look at the two-dimensional shadows of a cube, you see that the shape of the shadow depends on the placement of the light source. Similarly, the shape of a three-dimensional projection of a four-dimensional object depends on the vantage point from which you are looking at it. If you look at a cube directly from above the middle of one of its faces, you get a picture of a square that has a smaller square inside it. The two squares are connected to each other by lines (edges) that connect the corners. This type of a projection is called a Schlegel diagram. In the Schlegel diagram of a cube, you can see every face, vertex, and edge.

The bigger one of the two objects that form our piece is a three-dimensional Schlegel diagram of the 24-cell. There, you can see all 24 octahedral cells, with smaller ones placed inside a big one. It is made of aluminium tubes, connected together with ropes that run inside the tubes, and it is hang over the courtyard. The piece and the technique used to build it is inspired by a traditional Finnish Christmas decoration object, himmeli.

There are many ways to project a three-dimensional object on a two-dimensional surface (or a four-dimensional object in three-dimensional space). For instance, when maps are made, the surface of Earth – a sphere – needs to be projected on a two-dimensional plane. One way to do this is the stereographic projection. To do it, you imagine that the south pole of the sphere is placed on the plane. When you want to know where a given point on the surface of the sphere is projected, you draw a line from the north pole through the point. The line meets the plane at a point that will be the projection of the original point.

If you place a cube inside a sphere, inflate it so that its faces are on the surface of the sphere, and then do the above projection, you get a stereographic projection of a cube. The advance of such a projection is that you get a more holistic picture of the whole object.

The smaller part of our piece is a stereographic projection of the 24-cell. It is made of bent copper tubes that are connected with 3D-printed plastic pieces. A lamp is attached to the object so that the three-dimensional “shadow” of the four-dimensional 24-cell creates two-dimensional shadows on the walls.

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