What is the standard n dimensional unit cube?

1. What is the standard n dimensional unit cube? • Vertices: Points (e1,e2,…,en) where each ei is either 0 or 1. • Edges: The line segments connecting vertices that are a distance 1 away from of each other (i.e. differ in one entry).

2. 4 Cube Vertices and Edges

3. 5 Cube Vertices and Edges )

4. Questions… • How many vertices are in the standard n dimensional unit cube? • How many edges are in the standard n dimensional unit cube?

5. General Strategy for Sketching an n Dimensional Cube • Create the vertices. • Create an orthonormal basis for the view plane. These vectors will determine the coordinate axes in the view plane. • Create the coordinates for the projected vertices (in the view plane). • Determine the connection order. • Sketch the edges by connecting the appropriate points in the view plane.

6. A Convenient View Plane

7. 4 Dimensional Cube The sketch below shows the standard 4 dimensional unit cube using orthogonal projection onto the convenient view plane. The slides that follow show 5, 6, 7 and 8.

8. 5 Dimensional Cube

9. 6 Dimensional Cube

10. 7 Dimensional Cube

11. 8 Dimensional Cube