First of all, we all know that the length from the center of the sphere to any point on the sphere is the same. Then, we choose an axis that passes through the center of the sphere (through the North Pole and the South Pole). If we cut the sphere along the axis like a watermelon petal, we will get the outline of the meridian. If the spherical surface is bisected to the axis, you will get many circles, which are called latitudes. Therefore, the whole sphere is covered by two clusters of network curves of latitude and longitude, and every point on our earth can be determined by the value of latitude and longitude. Therefore, we can say that the sphere is two-dimensional, which mathematicians usually call S2.
Suppose we leave the earth for space now. In order to know our position accurately, we need to use three numerical values, namely longitude, latitude and height above the earth. Because we need three numbers to determine the position of outer space, we say that space is three-dimensional.
Now, let's talk about how to draw a map. One way is to project the earth onto a plane. First, you can choose a city, such as "Dakar", and then draw a straight line connecting the North Pole and the city. This straight line passes through another point on the desktop, which is called the projection of the city "Dakar".
Any point on the earth can be projected onto the desktop. The closer these points are to the North Pole, the farther their projections are, while the North Pole has no projection. In other words, its projection is at infinity, and the whole earth except the North Pole can be shown on the desktop. This map that appears on the desktop is called spherical projection.
But this projection didn't keep its original size. For example, South America is very small compared with North America.
In order to make it easier for everyone to understand this projection, we let the earth roll and always project from the highest point to the desktop, so we see that the projection of the mainland will gradually enlarge first and then gradually shrink, but in fact their "shape" has not changed. The "shape" here does not mean the exact shape, but the relationship between the points before and after the mapping has not changed, but the length has changed, so we say that spherical projection is a kind of projection.
If we expand the latitude and longitude again, when the earth rolls, we can see that they are always projected into circles or straight lines. Therefore, spherical projection can transform circles drawn on a sphere into circles drawn on a plane, and those circles passing through the highest point will become some straight lines.
Now let's move the ball and watch the same action from the bottom of the table. We can see that these longitude and latitude lines form two clusters of circles, and all the longitude lines meet at the South Pole and the North Pole.
This is the end of our first step towards four-dimensional space.