Exploring Hypercubes and Multi-Dimensional Coordinates, Exercises of Geometry

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Hypercubes

▪ This activity will explore cubes in more than 3 dimensions. ▪ We will refer to all shapes as cubes regardless of dimension – so a 2D cube (square) will still be called a cube.

Coordinates

▪ Every time we add a dimension, we add an extra digit in to our coordinates. ▪ What are the coordinates of the vertices of a 4D cube? ▪ 5D? ▪ How many coordinates would you write down for 6 dimensions? 7? 10? n?

Drawing in 4 dimensions

▪ 0 dimensions, a cube is a point ▪ We can then extend in 1 dimension to make a line. ▪ Can you describe how to move the line to make a square? ▪ Can you describe how to move the square to make a cube?

Drawing in 4 dimensions

▪ 3D to 4D? ▪ From a 3D cube, to draw a 4D cube, often called a Tesseract, take the cube and extend it along the 4 th dimension. ▪ We clearly can’t do this, but we can try! ▪ Method 1: Draw a cube, then ‘slide it’, connecting the corresponding vertices. ▪ Method 2: Draw a bigger cube around a smaller cube and connect the vertices.

Shapes of cubes

Dimension Vertices Edges Faces 0 1 0 1 1 2 1 1 2 4 4 2 3 4 In dimensions less than 3, one of the faces is the space around the object. Can you complete the table for 3 and 4 dimensions? Do you know a formula that connects vertices, edges, and faces. Does it work past 3D?

Finding faces

▪ This is a representation of a 4D cube. ▪ Can you find the 8 3D cube faces in this drawing?

Cubes inside cubes

▪ To simplify vocabulary, we will use the dimension of cubes to describe shapes. ▪ Complete the table, what patterns can you find? ▪ Can you explain the patterns? 0D cubes (vertices) 1D cubes (lines) 2D cubes (squares) 3D cubes 4D cubes Dimension 0 1 0 0 0 0 1 2 1 0 0 0 2 4 4 1 0 0 3 4 5