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Yi Taiyub
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[math] Glome in colors
Yi Taiyub at 2018-09-02 19:40
URL http://kallery.net/s.php?i=864

What is the shape of the universe?
Does it extend infinitely?
We can go around the earth forever but can not find the end. Likewise, the universe may be finite and endless.
Let's consider a mathematical model suitable for such a space.

A summary of Euclidean geometry is as follows:
A space is composed of a set of points and the relationship between them. The point is a location in space. Between two points, there are infinitely many other points. A line is a set of points in a specific direction at a particular point in space.  A plane is a set of points in all directions normal to a specific direction at a particular point in space.
The dimension of a space is the number of independent coordinates needed to locate a point in the space. The n-sphere is a set of points at a specified distance from a particular point in (n+1)-dimensional space. The n-ball is a set of points whose distance from a particular point is less than a specified length in n-dimensional space. 



In a one-dimensional space, there is only one direction. So it is straight. If a one-dimensional space is looped, the relationship between points can be represented by a closed curve. But the curve shows just the relationship between points, not the shape of the space. The shape of a one-dimensional space is a straight line.
A circle is a closed curve. So it can be a representation of the one-dimensional looped space.

On a sphere, if you go straight in any direction at any point, you will follow a great circle which can be a representation of a one-dimensional looped space. So a sphere can be a representation of a two-dimensional looped space.
The sphere shows just the relationship between points, not the shape of the space. The shape of a two-dimensional space is a plane.

Can a glome represent a three-dimensional looped space, just as a circle represents a one-dimensional looped space and a sphere represents a two-dimensional looped space?

A glome is in a four-dimensional space. Therefore we can not imagine its shape. So we replace the fourth dimension with the color value and assign two color values to each point except the points at the boundary of the glome. The two values for each point are the same absolute value but opposite in direction.



All paths are reflected at the boundary of the glome making the space bounded without a boundary. Every path passing through any point in the glome is continuous at the point even at the boundary of the glome.
This is analogous to the projection of a sphere.



In the projection of a sphere, each point on the disc except the circumference represents two points. One is a point on the upper hemisphere and the other is a point on the lower hemisphere. Likewise, each point in the glome except the surface represents two points. One is a point in the negative hemiglome and the other is a point in the positive hemiglome.
The two hemiglomes are separate spaces though they are represented by one globe in a three-dimensional space. This is the same as if the disc in the projection of a sphere represents two hemispheres.
Going straight on a sphere is transformed to following an ellipse on a plane in the projection of a sphere. And the movement from one hemisphere to another is represented by the contact between the ellipse and the circumference of the disc.




In a glome, the section where z is zero is the projection of a sphere.



Going straight is represented by following an ellipse on the great section and the movement from one hemiglome to another is represented by the contact between the ellipse and the circumference of the great section.
For every direction at any point in a glome, there is a plane containing the point, the direction and the origin. By cutting the glome with the plane we get a great section mentioned above. So in any direction at any point in a glome we go straight and get to the starting point. This means that a glome can be a representation of a three-dimensional looped space. But the glome shows just the relationship between points, not the shape of the space. The shape of a three-dimensional space is a space.
The distance, area and volume in a glome is different from those in Euclidean geometry.









A circular model has two dimensions, but the space represented by the model has only one dimension. The spheric model has three dimensions, but there are only two dimensions in the space that the model represents. Likewise, the space represented by the glome model does not have four dimensions. One more dimension is needed to make the space looped.
In general, if n-1 dimensions are zero we get a circle, the one-dimensional sphere, as a great section from the n-dimensional sphere. If n-2 dimensions are zero we get a sphere, the two-dimensional sphere, and if n-3 dimensions are zero we get a glome, the three-dimensional sphere.





For any two points in a glome, there is a plane containing the two points and the origin. By cutting the glome with the plane we get a great section where we can draw a great ellipse passing through the two points. This means that at any point in a glome we can get to any other point in the glome by going straight.
The shrinking of a loop is the process by which all points in the loop follow straight paths to a specific point. All straight paths towards the specific point intersect only at two points, the specific point and the opposite point of the specific point. Therefore, the order of all points on a loop is not changed during the shrinking.
This means that every loop can be continuously shrunk to a point in a glome.







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