Dimensions Exploration
Objective: Get a better understanding of dimensions.
We can add dimensions to a figure by extending the figure into the next dimension.
If we take a 0-dimensional shape (that is, a point), we can copy that shape and connect the vertices. This gives a line segment, and that is a 1-dimensional object.
Similarly, we can start with a 1-dimensional object, draw two copies of it and connect the vertices to create a 2-dimensional object.
Similarly we can take two copies of the two dimensional object, connect the associated vertices and create a 3-dimensional object.
1. Use this method of construction to create a 3-dimensional object (it should look like a cube).
2. Use this method of construction to create a 4-dimensional object.
3. Complete the following table:
| vertices | edges | faces | cubes | |
|---|---|---|---|---|
| 0-dimensional cube | 1 | 0 | 0 | 0 |
| 1-dimensional cube | 2 | 1 | 0 | 0 |
| 2-dimensional cube | 4 | 4 | 1 | 0 |
| 3-dimensional cube | ||||
| 4-dimensional cube |
4. Let V(n) be the number of vertices in an n-dimensional cube. Write down a formula for V(n) in terms of n. (n≥0)
5. Let E(n) be the number of edges in an n-dimensional cube.Write down a formula for E(n). I will leave it up to you if you want to write E(n) in terms of just n or if you write E(n) recursively in terms of E(n-1) and V(n-1). (n≥1)
6. Let F(n) be the number of faces in an n-dimensional cube.Write down a formula for F(n). I will leave it up to you if you want to write F(n) in terms of just n or if you write F(n) recursively in terms of F(n-1) and E(n-1). (n≥2)
7. Let C(n) be the number of cubes in an n-dimensional cube.Write down a formula for C(n). I will leave it up to you if you want to write E(n) in terms of just n or if you write C(n) recursively in terms of C(n-1) and F(n-1). (n≥3)
8. Use your formulas to predict the number of vertices, edges, faces and cubes in a 5-dimensional cube.
Handin: A sheet with answers to all questions.
