Chapter 8

THE TRIO in the mix.

Assume that one of the vertices of the hyper cube is (0,0,0,0). Is it true for hyper cubes that if you draw a line that starts at the origin and is 45 degrees from one of the axes it will intersect a vertex? If you take all of the possible lines that start at the origin and are 45 degrees from an axis and the lines that form the axis will you intersect all the vertices of the hyper cube?

If we connect two perpendicular axes in a 2D system with a line we form a 1D object (a line). If we connect the axis in a 3D system with lines we get a 2D shape (a triangle). Is it true that if we connect the axes in a 4D coordinate system we get a 3D shape? We envision this to be similar to the concept of duality.

Polar coordinate: In 2D we need a length and an angle. In 3D we need a length and two angles. Does it hold that in 4D we need a length and three angles? Why? Can one say that the system to locate a point on earth (longitude and latitude) is very similar to polar coordinates.

If one has the vector (x,y) and one brings it into 3D one would just define the value of the new axis to be 0 (x,y,0). This would give us the exact same vector as in 2D. How can one bring a vector down from a higher dimension to a lower and still preserve the orginal?

If one multiplies a complex vector by i, the resulting vector would be the same but rotated 45 degrees counter clock wise. If one multiplied it by i^2 , one would rotate the vector 180 degrees. If one multiplies the vector by -i, it would rotate the vector 45 degrees clock wise. The complex coordinate system seems very like the polar coordinate system.

Exercise: Explain the math behind a projection, a transformation, and a rotation using matrices. Why is it true that to undo a projection, a transformation, or a rotation one must multiply by the inverse of the original matrix?

Prof. B. response