Enjoy.
A Chebyshev hexagon is simply an hexagon in Chebyshev space. For purposes of this discussion, we’ll deal only with regular polygons.
Essentially that’s all there is to them. Just your everyday regular hexagon, that by some twist of fate happens to be in Chebyshev space.
Well in case you’re wondering, what Chebyshev space is, Chebyshev space is the maximum metric (and if you’re wondering what a metric or space, or even maximum is, then this is the point where I say in an almost angry tone: “GOOGLE IT :) :)” ).
It is defined over a vector space, where the distance between two points, is the value of the largest separation along any coordinate.
Technically, it’s defined as:
which means that for a n-dimensional vector space (n number of coordinates), the distance between any two points p and q is given by the maximum of the differences between their individual coordinates.
So I’ll explain with two examples.
1. Let’s assume Heaven is a 5 dimensional Chebyshev space (maybe that’s why prophets and people that see ‘visions’ say it’s indescribable).
and I’m at somewhere say, St. Peter’s square with coordinates (4, 15, 6.8,7,7) and an angel is at (6,8,10.5,10,7). The difference between the two points is (2,7,3.7,3,0). In Chebyshev space, the distance between two points is the largest individual difference. Hence the distance between Me and the angel is 8 heavenly units.
2.Coming down to earth, literally, let’s look at the distance between my room in Blk 1, rm 108 and Tolu’s room in Blk 4, 313 Faj (if you’re not an Ife student and you don’t know what Faj is, well that’s tough luck for you). There are three dimensions in this scenario They are:
1. Up/Down: I stay in floor one, and he stays in Medical floor. There is a vertical displacement component.
2. Front/Back: Block one is in front of block four so there’s like a north/south horizontal displacement between them.
3. Left/Right: Facing front, block one is to the right of block four, so there’s like a horizontal east/west displacement.
So in normal space(Euclidean space) the distance between my room and Tolu’s is the square root of the sum of squares, but since we’re considering Chebyshev, it’s going to be the largest of the three differences. In this case, it’s likely the 3rd coordinate. That means if Faj was a Chebyshev space, the distance between my room and Tolu’s room would be equal to just the east/west horizontal distance between my room and Tolu’s room.
I guess with this. we get a general idea of how Chebyshev space works.
Now a Chebyshev hexagon is just an hexagon that has all the properties of a regular hexagon. The only difference being that the Chebyshev hexagon is in Chebyshev space.
I go forward to explain my line of thought below.
A regular hexagon (or any other polygon for that matter), in any space, is just a shape that divides the circumference of a circle in that space, into an equal number of segments, so for an hexagon, it’s six segments.(This definition is not official. I created it, so believe it at your risk)
Therefore, a Chebyshev hexagon, is a shape that divides a Chebyshev circle, which is actually a square (yep, you read it right. For some weird reason, that might take too much space to explain, a Chebyshev circle happens to be a square, read the wiki page to learn about this great deep magic).
Therefore, a Chebyshev hexagon is a six pointed shape that divides a square into six equal segments. But an important point to take note of, is that our distance is going to be measured in Chebyshev space, and not Euclidean(normal) space.
Haven’t settled that, the model for the hexagon is described below.
A Chebyshev circle of radius R, is actually a square of side 2R. The aim is to divide this circle(square) into six equal(in chebyshev distance) sides.
Now, since an hexagon has an even number of shapes, it must possess symmetry of some sort.
The image is displayed below.
The hexagon is shown in the figure above(In the red lines). It is a regular hexagon (meaning it has equal lengths and angles), with side length R(which is also the radius of the circle, or square).
On inspection, the sides appear to be unequal, but upon analysis, we see that they are indeed equal.
Since we are in Chebyshev space, the length of each side is the larger of the difference between its x and y coordinates. And for each side this can be seen to be R.
So there you have it, A Chebyshev hexagon- from imagination to construction.
Hope it was worth the while.
Thanks :) :)
Soikramari D. Bestman.
