## Imaginary Numbers as coordinates

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### Imaginary Numbers as coordinates

One really weird property about imaginary numbers is that if they are used as values in coordinates they can produce two different points in which the distance between the coordinates is 0.

For instance if you have the points (i, 0), (0, 1) and you use the distance formula to figure out the distance then you get
d=sqrt((i-0)2+(0-1)2)=sqrt((i)2+(-1)2)=sqrt(-1+1)=sqrt(0)=0 so even though the points have different x and y values in their coordinates they have a zero distance.

Also if you have a right triangle with the lengths of the sides 1 and i then if you use the Pythagorean Theorem you get 12+i2=c2
1-1=c2 0=c2 c=0 so that the legs of the right triangle are longer than its legs.
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anderscolingustafson
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### Re: Imaginary Numbers as coordinates

The way Coxeter (Regular Complex Polyhedra) finds the distance for coordinates, is not a^2, but aa*, where a* is the conjucate of a. This eliminates any preferred direction of axis.
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### Re: Imaginary Numbers as coordinates

And just to be clear: if a = x+yi, then a* = x-yi. So aa* = x^2 + y^2.
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### Re: Imaginary Numbers as coordinates

Why would we actually need imaginary coordinates? Can't we represent (i,0) simply as (0,1,0,0) since even otherwise we shall eliminate any preference of direction?
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### Re: Imaginary Numbers as coordinates

Because they allow us to define "complex polytopes", that is, polytopes whose vertex coordinates are complex numbers. These things do not correspond with any Euclidean polytopes, but they do include incidence structures that are not covered by Euclidean polytopes. One example is the complex hexagon octagon, which has an edge structure equivalent to the edge structure of a tesseract (but the higher order faces of the tesseract have no correspondending elements in the complex hexagon octagon).

Complex polytopes exist in complex spaces, where vectors may have complex coordinates. These are incredibly hard to visualize, but they do have applications in subatomic physics. For example, the shape of electron orbitals corresponds with the real part of a larger manifold that exists in complex 3D space (i.e., effectively 6D). You could imagine that as being the intersection of 3D real space with a 3D complex manifold. The d-orbitals and f-orbitals, for example, appear to have more than one possible shape, but actually, they are all congruent to each other in 3D complex space; it's just that in 3D real space we see a different "slice" of them. The "odd" shape of the d orbital, for instance, is merely a consequence of it having a different orientation with respect to 3D real space. In its "native" 3D complex space, it actually has exactly the same shape as the other d orbitals.
Last edited by quickfur on Fri Jan 16, 2015 7:20 pm, edited 1 time in total.
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### Re: Imaginary Numbers as coordinates

Complex polytopes may not correspond with Euclidean polytopes, but then would it be right to say that a complex n-D space is effectively a real 2n-D space?

If what I understood is right, it must be easy for us to visualise complex 1-D space. How different would a complex line segment be from a Euclidean 2D figure?
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### Re: Imaginary Numbers as coordinates

The problem with conflating complex n-D space with Euclidean (2n)-D space is that you're changing the interpretation of the objects. For example, in Euclidean 2D space, polygons have well-defined inside and outside regions, edges have well-defined interiors and exteriors (points on the line segment that defines the edge, vs. points outside), and a chain of connected edges form a circuit that inscribes a polygon. However, these concepts are undefined in complex space. A complex line segment, while on the surface it may look like an Euclidean polygon, does not have an interior or exterior, because points on the complex line are unordered. As a result, edges may have more than 2 bounding points, and faces are not contrained by their circuit of edges, but encompass an entire plane. Complex polytopes therefore do not partition space -- there is no such thing as "inside" a complex polytope or "outside" a complex polytope. As a result, there's no such thing as the distinction between convex and non-convex complex polytopes. Many geometric intuitions from Euclidean space do not hold in complex space. That's why visualizing a complex polytope is a lot harder than it might seem at first glance!

Btw, I made a mistake, it's not the complex hexagon, but the complex octagon, that has an edge structure resembling a tesseract. However, that's just one interpretation. There are also other interpretations that have no correspondence with any Euclidean polytope at all.
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### Re: Imaginary Numbers as coordinates

Isn't it that complex vertices correspond to euclidean vertices of twice the dimensional space,
but that then complex edges should correspond to euclidean faces, not to edges again!?

And in general, n-dimensional elements of some complex polytope correspond to 2n-dimensional
elements of its representation within euclidean space (likewise of twice the dimension).

That is, the odd-dimensional elements of the euclidean space representation do not occur
within the incidence structure of those complex polytopes.

Btw. just as the euclidean edge is the convex hull of the 2 spanning vertices, a complex edge
might be "seen" as the (euclidean 2D) convex hull of the spanning vertices too.

It generally might be a good idea to find the corresponding euclidean analogues for all those
complex polytopes (which then surely have a higher structure, because of adding the odd-dimensional
elements). Those then are easier to grasp, even if higher dimensional, as those complex ones.
And thus might provide insides onto those then too.

--- rk
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### Re: Imaginary Numbers as coordinates

Using complex numbers as coordinates is an ideal way to introduce clifford rotations, and the various regular complex polytopes are a good introduction to some of the subsymmetries of the regular polytopes.

If you take the line y = Ax + B, it is easy to show some interesting things in 4D. For example, where B=0, there is a series of points x,y where wy = A wx.

Put w = cis(vt) where v is a velocity and t is a time, and you can rotate the whole of 4-space where the point follows a path where A never changes. That is, one can have rigid rotation of all of space around a point. It also means for a set swirly space, two points define an argand diagram.

If you now plot the coordinates of A as X+iY, and set up a sphere whose diameter is (0,0,0) to (0,0,1) [a diameter-1 sphere sitting on 0 on the argand diagram], you get the sphere of rotations from where the line from (X,Y,0) to (0,0,1) passes through this sphere. When we say the decagons of the (3,3,5) are in icosahedral arrangment, they actually produce the vertices of an icosahedron on this sphere. Likewise, the order-six swirls produce the 20 vertices of a dodecahedron, and the 30 vertices of an icosadodeca make the 30 order 4 rotations.

The complex polygons like 3(5)3 and 5(3)5, are swirl-rotations of order 3 and 5 on top of the poincare dodecahedron. All of these are various kinds of swirl symmetries.

One should note that where the two outside numbers are different, ie p(2q)r, where p,q,r is one of 2,3,3, or 2,3,4 or 2,3,5, that the resulting thing is not a subgroup of any of the regular polyhedra, because if the numbers on either side (ie p\= r ) then the class doubles.
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