projecting a 4-dimensional hypercube onto a hyperplane
Budget: $2 – $8 USD
This task involves projecting a 4-dimensional hypercube onto a hyperplane (ie a 3-dimensional space).
Let the hypercube have its vertices at the points (±1,±1,±1,±1), where all 16 vertices are obtained by using all combinations of plus and minus signs. The hyperplane to be projected must have the fourth dimension coordinate 0, in analogy with 3D, the xy plane is the plane where the third dimension coordinate z = 0.
The projection must be a so-called oblique projection along a vector U from the vertices to a point in the hyperplane (ie the 3-dimensional regular space). One way to solve it is to go from a corner of the hypercube along the vector U until you reach the hyperplane (the one with the fourth component equal to zero), or create a mapping matrix which is a four-dimensional oblique projection to the right hyperplane along U (this can be very hard to find, I'm not sure if there is described in any book how to easily calculate such a matrix but it
a simple formula exists).
Use the vector U = (1, 3, 2, 7).
Report the 3-dimensional coordinates that the vertices of the hypercube get after the projection and draw the projected hypercube in a figure with edges between nearby vertices. Remember that there are no edges between all the vertices (use the analogy of a regular cube and figure out between which vertices of the cube there are edges).
Let the hypercube have its vertices at the points (±1,±1,±1,±1), where all 16 vertices are obtained by using all combinations of plus and minus signs. The hyperplane to be projected must have the fourth dimension coordinate 0, in analogy with 3D, the xy plane is the plane where the third dimension coordinate z = 0.
The projection must be a so-called oblique projection along a vector U from the vertices to a point in the hyperplane (ie the 3-dimensional regular space). One way to solve it is to go from a corner of the hypercube along the vector U until you reach the hyperplane (the one with the fourth component equal to zero), or create a mapping matrix which is a four-dimensional oblique projection to the right hyperplane along U (this can be very hard to find, I'm not sure if there is described in any book how to easily calculate such a matrix but it
a simple formula exists).
Use the vector U = (1, 3, 2, 7).
Report the 3-dimensional coordinates that the vertices of the hypercube get after the projection and draw the projected hypercube in a figure with edges between nearby vertices. Remember that there are no edges between all the vertices (use the analogy of a regular cube and figure out between which vertices of the cube there are edges).