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[math] The glome as a model of the loopace
Yi Taiyub at 2018-09-02 19:40

A space is composed of a set of points and relationships between them.
The point is a location in space. Between any two points, there are infinitely many other points.
A line is a set of points in a particular direction at a particular point in space.
A plane is a set of points in all directions normal to a particular direction at a particular point in space.

To get to a point in the space, you have to travel half way to that point and half way to that halfway point and so on. There's no end to the halfway so you can't reach the point.
This paradox applies to all changes that are assumed to be continuous and the change of time is not an exception. To overcome this paradox, we assume that time and space are discrete and composed of elementary units that cannot be divided any more.
We also assume that the size of the unit space is not greater than the smallest particle in the universe and the length of the unit time is not longer than the time the fastest particle in the universe stays in one unit space.
This assumption prevents multiple particles from coexisting in one unit space at the same time and prevents one particle from being in multiple unit spaces at the same unit time.

We can go around the globe, but we can not find the end. The universe can be the same as the earth's surface but three-dimensional instead of two-dimensional. Then it would be endless and finite.
Suppose a looped space (for short, loopace) as follows:

At any point in the space, if you go straight in any possible direction, you will return to where you leave off.

In a one-dimensional space, there is only one direction. So there is only one straight path too. If the 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 the one-dimensional loopace is a straight line.
A circle is a closed curve. So it can be a representation of the one-dimensional loopace.
x^2 + y^2 = r^2

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 loopace. So a sphere can be a representation of a two-dimensional loopace. We can call it a two-dimensional circle.
x^2 + y^2 + z^2 = r^2
The sphere shows just the relationship between points, not the shape of the space. The shape of a two-dimensional loopace is a plane.
In a two-dimensional space, there are two directions perpendicular to each other at any point. If the space is looped the path lengths along the two directions are finite. So the area is finite too because it is not greater than the maximum product of the two finite lengths.
If a two-dimensional space has a boundary, there must exist a line tangent to the boundary on a certain point of the boundary. The point is the endpoint of a path along a direction perpendicular to the tangent line. This contradicts the definition of a loopace. So the two-dimensional loopace has no boundary.

In a three-dimensional space, there are three directions perpendicular to each other at any point. If the space is looped, the path lengths along the three directions are finite. This induces that the volume of the space is finite too because it is not greater than the maximum product of the three finite lengths.
If a three-dimensional space has a surface, there must exist a plane tangent to the surface on a certain point of the surface. The point is the end point of a path along a direction normal to the tangent plane. This contradicts the definition of a loopace. So a three-dimensional loopace has no surface.
Like a circle and a sphere, can a glome be a representation of a three-dimensional loopace?
x^2 + y^2 + z^2 + w^2 = r^2

A one-dimensional loopace can be drawn in a two-dimensional plane. A two-dimensional loopace can be drawn in a three-dimensional space. Then a three-dimensional loopace may be drawn in a four-dimensional space. But we cannot imagine a four-dimensional space. So we replace the fourth dimension with color values.

In the projection of a sphere, each point in the disc 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 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.

12 straight paths passing through a point on a sphere are projected to the xy plane.

In a glome, the section where z is zero is a sphere.
x^2 + y^2 + w^2 = r^2

It is like 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 loopace. But the glome shows just the relationship between points, not the shape of the space. The shape of a three-dimensional loopace is a space.
The great section of the great section of the glome is a great circle. So we can call the glome a three-dimensional circle.

For given two points, any point within both the radii from the two points can be the origin of a glome. So there can be infinitely many straight paths between the two points. But, of course, there is only one straight path between the two points in the loopace.

The second dimension in a circular model and the third dimension in a spherical model do not exist in the space which the models represent. In the same way, the fourth dimension in a glome model does not exist. It just interconnects other dimensions to make the space looped.
In general, if n-1 dimensions are zero we get a circle, the one-dimensional circle, as a section from the n-dimensional circle. If n-2 dimensions are zero we get a sphere, the two-dimensional circle, and if n-3 dimensions are zero we get a glome, the three-dimensional circle.

50,000 points in a glome

300 straight paths passing through a point in a glome

Like a sphere, every loop can also be continuously tightened to a point in the glome.

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