If the universe is spread out in 4D space, & there is a system like the solar system, & there is a planet with civilization like the earth, how do day & night & the seasons look like? I would like to understand somehow by working with analogies. First, we will briefly discuss the circle & sphere.
Circle
Circle: a closed curve every point of which is equally distant from a fixed point within it.
On the circle, one can move in only one direction; thus, the location on the circle is determined by one angular parameter between 0° & 360°.
The point of \(\theta\) degrees is on \((\cos \theta, \sin \theta)\).
Sphere
Sphere: a solid figure so shaped that every point on its surface is an equal distance from the center.
On the sphere surface, one can move in two directions: north-south & east-west. Thus, the location is determined by two angular parameters: latitude & longitude.
Pole: either end of an axis, especially on the earth.
Equator: the mid-section; an imaginary circle around the earth that is everywhere equally distant from the two poles.
Latitude: angular distance north or south from the earth's equator measured in degrees.
Longitude: angular distance usually expressed in degrees east or west from the prime meridian thru Greenwich, England.
Earth rotates along the z-axis. Equator is on xy-plane.
The direction of rotation on any point (except for the poles) is called east, & the opposite direction is called west. The speed of rotation varies from planet to planet.
A group of points where the latitude is \(Lat\) forms a circle with the radius \(\cos Lat\). It's on the plane \(z= \sin Lat\).
Thus, the point of \((Lat, Lon)\) is on \((\cos Lat \cos Lon, \cos Lat \sin Lon, \sin Lat)\).
Glome
In 4D space, the equivalent of the sphere is called the hypersphere, or the glome. From now on, all we can do is to imagine. It's hard to imagine it, but it's just all points same distance away from the center, extending to the fourth dimension.
On the glome surface, one can move in three directions; thus, three angular parameters show ur location. There are two separate models which explain this.
The "traditional" model
Pole: either end of an axis.
Equator: the mid-section; an imaginary sphere around the planet that is everywhere equally distant from the two poles.
Hypernorth: The direction to the North Pole. The opposite is hypersouth.
Three independent directions: north-south, east-west, hypernorth-hypersouth.
Hyperlatitude: angular distance hypernorth or hypersouth from the equator.
Latitude & longitude: a pair of angular values which show a location on a sphere.
Hyperspherical coordinate system:
Hypernorth Pole is on \((0,0,0,+1)\). Hyperlatitude is +90°.
Hypersouth Pole is on \((0,0,0,-1)\). Hyperlatitude is -90°.
Equator is on xyz-hyperplane.
Stereographic projection:
Just like the surface of the sphere can be projected into a 2D map, we can peoject the "surcell" (3D surface) of the glome into a 3D map. In the diagram above, the Hypernorth Pole is in the center, while the Hypersouth Pole is on the point infinitely away from the center.
Note that the stereographic projection shows only the "surcell" (3D surface) of the hypersphere (glome), & the labeled coordinate points are the intersections of the coordinate axes with the surcell of the hypersphere; the coordinate origin is at the center of the hypersphere, & is not represented in the projection.
The "toroidal" model
I apply this in my model of the 4D planet, as it brings more uniqueness & diversity of the 4D world.
The planet has two equators. One is on xy-plane; the other is on zu-plane. I call them South Track & North Track.
The planet has no poles. However, South Track & North Track work just like poles in 3D.
Yin-yang: New terms for directions in 4D space. Can be changed if you're not happy with them.
Three independent directions: north-south, east-west, yin-yang.
From a point on a 4D planet, north is the direction to the closest point on the North Track (R in the figure below). South is the direction to the closest point on the South Track (Q in the figure below) & opposite of north.
(Yellow: east-west; pink: north-south; cyan: yin-yang)
Latitude: angular distance north or south from the earth's equators. It has a range of only 90 degrees.
Longitude: a pair of angular values, east-west & yin-yang from the prime meridian. Each has a range of 360 degrees. Just like in 3D, it is necessary to artificially determine which point is the 0° of east-west & yin-yang.
Hopf coordinate system:
\(\eta\) is latitude; \(\xi_{1}\) is longitude; \(\xi_{2}\) is colongitude. \(\eta\) moves in a range of only 90°, whereas \(\xi_{1}\) & \(\xi_{2}\) move in a range of 360°.
South Track is on xy-plane. Latitude is 0°.
North Track is on zu-plane. Latitude is 90°.
Every point on the South Track is in the same distance from every point on the North Track, just like poles in 3D. The distance is a quarter lap.
Earth rotates along the xy-plane & zu-plane at the same time, called a double rotation. In my model of the 4D planet, the rotation on the xy-plane is faster than that on the zw-plane.
Note: If the speeds of the two simple rotations are identical, called isoclinic rotation, there are infinitely many ways to define equators.
A group of points where latitude is 45° lies halfway between the two Equators. It partitions a planet into two equal halves. It's in the shape of Clifford torus, which is made of two equal circles.
By east-west and yin-yang rotations, any point moves in the torus surface. A "flat torus" has the property of being able to expand into a rectangle without stretching, so we can draw it on paper as a two-dimensional map connected up-down & left-right, and see how the points rotate and move. Let's take a look!
- Double rotation (green) proceeds diagonally along the torus plane. If the angle is \((\alpha, \beta)\), while a point moves for \(\alpha\) in the east, it moves for \(\beta\) in the yang. If the ratio of angles is an irrational number, you will never return to the starting point no matter how many laps you do.
- Simple rotation (blue) has zero angle on one side, so it moves only in the east or yang direction. You will return to the original position after one lap.
- Isoclinic rotation (red) proceeds diagonally, but since both longitudes change by the same magnitude, it returns to the beginning in one lap.
The following image is the stereographic projection of the glome, visualizing some of the torus surfaces. Thin red lines show the paths of points which move as isoclinic rotations.
Move on to the 4D planet article.