In this contemplation of geometry, we will construct a tetrahedron from scratch while reflecting on how some of its features may be associated with Mount Olympos.
Before we can build Olympos, we have to start with empty space. Nothing. Nada. Zip. The Archetypal Newbie!
What's the first thing you can put in empty space? A single point. It's a dot with Zero dimensions and is just about useless by most standards. It's like an infinitely small circle.
That's 1 point, 0 dimensions.
And if you add a second point? Now you can see dimensional space, because with 2 points you can derive length, which is 1 dimension. You can even refer to a unit of length: the distance between the 2 points. But at this point everything is relative to the 2 points, since they're the only things that exist in the empty space. Still not much you can do, aside from exploring abstract equivalents of Zeno's paradox and stuff, like bisecting a line made between the two points, bisecting the line between the 1st point and the half-way point, etc.
2 points, 1 dimension. Line, distance/length.
In order to get to 2-dimensional space, you need a third point not placed along the line made by the first 2 points. Place this point anywhere at all in the empty space, and you are able to establish a flat plane of some kind. In fact, you can connect the points to make a triangle, no matter where you placed the three points. You can now refer in a more or less meaningful way to the length and width of the triangle. In 2-dimensional space, you can start thinking about relatively useful stuff.
3 points, 2 dimensions. Plane, distance/length, width.
Adding a fourth point not located on the plane established by the first 3 points finally gets us into 3-dimensional space. Length, width, and height may now be meaningfully discussed. You can now form a solid. Now we're getting somewhere.
4 points, 3 dimensions. Solid, distance/length, width, height.
The most basic kind of "regular object" you can make in 3-dimensional space is a tetrahedron. It uses the minimum number of points (4), equidistantly spaced from one another.
Now, at this point, some people might feel their brains oozing out their ears. That's perfectly okay (and probably a good start for most of us). If you're using a Java-enabled browser, you can look at this page that will let you play with tetrahedrons. It's pretty neat, and may prove helpful in getting past this point.
What are some of the basic properties of a tetrahedron? Quite a few, actually, so I'll stick with a few basic details. Each side of the tetrahedron borders *every other side* of the tetrahedron. This is an unusual property.
They have 4 sides, each of which is a regular triangle. In other words, a tetrahedron consists of four 3-sided sides. There are exactly 12 angles on a tetrahedron. Each of the 4 sides has 3 of the angles, and all angles are identical in their dimensions.
If you want an interesting image running through your head, notice that the shape of a tetrahedron is the basic shape of an archetypal mountain. If you "unfold" a tetrahedron, it unfolds into a triangle with the same basic dimensions as one of the tetrahedron's sides.
Imagine that one deity inhabits each of the 12 interior angles of the tetrahedron, and you can see a bit of a vision of Olympos with the full Dodekatheon in place! Now, isn't that neat?