Geometry is a relatively simple model, and is based on 3 fundamental presuppositions about the nature of the world:
1. The only component of nature is space.
2. Space exists in 3 dimensions.
3. In each dimension, space is infinite.
Let us examine these presuppositions more closely:
1. The only component of nature is space:
It shall be accepted as given that it is meaningful to consider space completely isolated from time. In the 2,300 years since Euclid wrote the Elements, the most influential book in the history of science, this is the only one of these presuppositions that has been reexamined and successfully challenged, and this presupposition has evolved multiple times over this period. More recent models of nature have come to include time as well as space, and most recently time integrated with space, as space-time. In other words, although more recent models of nature presuppose 2 components of nature, space and time, geometry presupposes 1. With no representation of time in this model, there can be no representation of motion in space, as motion in space requires time. Therefore, this model at best represents only existence and position in space.
2. Space exists in 3 dimensions:
It shall be accepted as given that humans are aware of 3 distinctions, or extensions, in space, known as dimensions. Modern theories sometimes claim more, such as 10, or 11, or 25 dimensions, but these are still controversial, and anyway are beyond human awareness. Each of these dimensions represents extension in a pair of directions in space that can be referred to by such names as length, width, and height (or depth). All of space always exists in all 3 dimensions.
3. In each dimension, space is infinite:
Infinity takes on 2 forms, infinitely large and infinitely small.
Structural elements of Space
On the basis of these 3 fundamental presuppositions about the nature of nature, the following can be deduced about the structure of space:
1. There exists a set of fundamental elements that compose the structure of space.
2. Each of these elements exists in all 3 dimensions.
3. For each element, each of the 3 dimensions is infinite.
4. There exist exactly 4 such elements, wherein 0, 1, 2, or 3 of the infinities are large.
These are the implications about the structure of nature that are drawn by geometry on the basis of its presuppositions. According to geometry, the four building blocks, or elements, that compose space, are recognized to be as follows:
1. The first fundamental element of space is the point. The point exists in 3 dimensions. Each of the 3 dimensions is infinite. In a point, each of the 3 dimensions, length, width, and height, is infinitely small. Because 0 of the 3 dimensions are infinitely large, the point is called 0 dimensional, and it is understood that points exist in 0 dimensions.
2. The second fundamental element of space is the line. The line exists in 3 dimensions. Each of the 3 dimensions is infinite. In a line, width and height are infinitely small. Length, however, is infinitely large. Because 1 of the 3 dimensions is infinitely large, the line is called 1 dimensional, and it is understood that lines exist in 1 dimension.
3. The third fundamental element of space is the plane. The plane exists in 3 dimensions. Each of the 3 dimensions is infinite. In a plane, height is infinitely small. Length and width, however, are infinitely large. Because 2 of the 3 dimensions are infinitely large, the plane is called 2 dimensional, and it is understood that planes exist in 2 dimensions.
4. The fourth fundamental element of space is simply called space, or 3 dimensional space. The totality of space exists in 3 dimensions. Each of the 3 dimensions is infinite. In space, each of the 3 dimensions, length, width, and height, is infinitely large. Because all 3 of the 3 dimensions are infinitely large, space is called 3 dimensional, and it is understood that space exists in 3 dimensions.