hypercomplex grapher

The imaginary number i is defined as the square root of -1. But what if we change that? What if it's the square root of i itself, or of 1+i, or 3+2i?

This page is a exploration of graphs using hypercomplex numbers, systems that may have a different definition of i, as discussed in my video.

using shapes

The core idea of this page is to apply a function to a shape. A pentagon cubed, the sine of an octagon, the root of a spiral. What do these mean?

Each shape is made up of points in the plane. A square may include (1,0) and (0,-1) and many other points. We assign each point to a hypercomplex number. (0.3, 0.7) becomes 0.3 + 0.7i.

To apply a function to the shape, we simply apply the function to each hypercomplex number in the shape, and then graph all outputs. For example, if 0.3 + 0.7i is part of a shape, and we want to square that shape, then we calculate:

(0.3 + 0.7i)²

= 0.09 + 0.42i + 0.49i²

The resulting location depends on how we evaluate i².

defining i²

The typical complex numbers use i² = -1, but you can drag the i² icon around to try different definitions! We get a 2-dimensional system by setting i² equal to any a + bi.

However, there are only 3 distinct systems. If b² < -4a then the system is isomorphic (effectively the same) as the complex numbers. There will be some value that squares to -1.

If b² = -4a then the system is isomorphic to the the duals, where there is some nonzero ε that squares to 0.

And if b² > -4a, then the system is isomorphic to the split complex, where some nonreal j squares to 1.

The parabola in the background denotes this border. When i² is on the parabola we have the duals. When it is to the left we have the complex, and to the right we have the splits.

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