My own work involving the first action of creation involved hte creation of an effective tetrahedra to produce both SPACE and TIME. This led to an investigation of tetrahedra and now bwe have this work which is a huge clarification. An unexpected one even when you know what does not work at all.
So this is very welcome work.
Understand that we can not perfectly pack tetrahedra at all and it was not obvious why. This makes it much clearer.
All this is foundational to understanding particle physics.
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Why Triangles Are Easy and Tetrahedra Are Hard
The triangle angle sum theorem makes working with triangles easy. What happens when you can’t rely on it?
Olena Shmahalo for Quanta Magazine
January 31, 2022
https://www.quantamagazine.org/triangles-are-easy-tetrahedra-are-hard-20220131/
Do you think there’s a triangle whose angles measure 41, 76 and 63 degrees?
At first, answering this may seem easy. From geometry class we know that the sum of the measures of the interior angles of a triangle is 180 degrees, and since 41 + 76 + 63 = 180, the answer must be yes.
But there’s more to this question than meets the eye. The triangle angle sum theorem tells us that, given a triangle in flat Euclidean geometry, the sum of the measures of the interior angles is 180 degrees. But our problem doesn’t give us a triangle. Instead, we are asked if one exists. The triangle angle sum theorem doesn’t directly answer that question. But it can help us construct the triangle we want.
To satisfy the triangle angle sum theorem, each angle in a triangle has to be less than 180 degrees, which means we can always arrange two of them on the same side of a line segment. Let’s put the 41-degree angle and the 76-degree angle at either end of line segment AB, like this.
Samuel Velasco/Quanta Magazine
The rays extending from A and B can’t be parallel, because in Euclidean geometry this would require these “same-side interior” angles to be “supplementary” — that is, to sum to 180 degrees. These angles don’t, so the rays aren’t parallel, and if they aren’t parallel, they must intersect.
Call their intersection C. Now we’ve got a triangle, and now we can apply the triangle angle sum theorem. The third angle must measure 180 − (41 + 76) = 63, and so △ABC is exactly the triangle we were looking for.
This argument can be generalized to show that any three angle measures that sum to 180 degrees can make a triangle, and one immediate consequence is that it’s easy to find triangles whose angle measures (in degrees) are all rational numbers. Just start with any two positive rational numbers whose sum is less than 180; call them x and y. Then 180 − (x + y) is also a rational number, and since x + y + (180 − (x + y)) = 180, you can make a triangle with those three rational angles.
Quantized Academy
Patrick Honner, a nationally recognized high school teacher from Brooklyn, New York, introduces basic concepts from the latest mathematical research.
As easy as it is to make triangles with rational angles, a similar problem in three dimensions proved so challenging that it took decades for the world’s best mathematicians to resolve. What makes this kind of problem so much harder one dimension up? To understand this is to appreciate the triangle angle sum theorem even more.
The related problem in three dimensions involves tetrahedra — four-sided shapes with triangular faces. You could think of them as the three-dimensional version of triangles. In two dimensions the triangle is the simplest closed shape you can make with flat sides, and you need three line segments to do it. In three dimensions, the tetrahedron is the simplest closed shape you can make with flat sides, and it takes four triangular faces to do it.
The four triangular faces of a tetrahedron are like the three sides of the triangle. But how should we think about the angles? You could imagine a “solid angle” at each of the four vertices of the tetrahedron. But the question we’re interested in involves the “dihedral angles” formed by the intersecting faces.
If you sketch two intersecting planes, you’ll discover many different angles you could measure. Which one should you choose to represent the dihedral angle?
The answer is to rotate the intersecting faces until they look like a two-dimensional angle.
It’s this angle that we associate with the dihedral angle.
In a tetrahedron each of the four faces intersects every other face, making six edges and six dihedral angles. For decades, mathematicians wondered what kinds of tetrahedra have six rational dihedral angles. As before, an angle is considered rational if its degree measure is a rational number. This is equivalent to its radian measure being a rational multiple of π (To see the equivalence, note that to convert from degrees to radians you multiply the degree measure by π180∘, so if the degree measure is rational, then the radian measure is a rational multiple of π, and vice versa.)
We’ve seen how easy it is to make triangles with rational angles, but the problem is much trickier for tetrahedra. Consider the following simple tetrahedron OABC that is formed from a corner sliced off a cube.
Right away we see that three of the dihedral angles in this tetrahedron are right angles, since they were formed by the faces of the cube. It’s convenient to identify each dihedral angle with its edge, so in this tetrahedron the dihedral angles on OA, OB and OC are all right angles.
If you slice the cube so that OA = OB = OC, the dihedral angles at AB, AC and BC will all be congruent. Let’s cut the cube so that OA = OB = OC = 1, and then calculate the measure of the dihedral angle at BC. The key to measuring this dihedral angle is to draw segments from O and A to the midpoint of BC. Let’s call that point M.
If we rotate the tetrahedron to look at the dihedral angle at BC from the side, we’ll see ∠AMO, which has the same measure. To measure ∠AMO we need the lengths OA and OM. We already know that OA = 1, and to find OM we just need to take a closer look at triangle ΔOCB.
Since ∠BOC is a right angle, this is a right triangle, so we can use the Pythagorean theorem to find that BC = 2–√. And because M is the midpoint of BC, we know MC = 2√2. But in addition to being a right triangle, ΔOCB is also isosceles, because OB = OC. This makes it a 45-45-90 triangle, which means that ∠OBC and ∠OCB both measure 45 degrees. The fact that ΔOCB is isosceles guarantees that OM is perpendicular to BC, which makes ΔOMC a right triangle as well. But if the measure of ∠OMC = 90° and the measure of ∠OCB = 45°, then the triangle angle sum theorem tells us that the measure of ∠MOC = 45°. That makes the smaller triangle ΔOMC isosceles, so OM= MC = 2√2.
Now we’re finally ready to find the measure of ∠AMO.
In ΔAMO we know AO = 1 and OM = 2√2, and since ∠AOM is a right angle, we can apply trigonometry. In a right triangle the tangent of an angle is the ratio of the opposite side and the adjacent side:
tan∠AMO = 12√2 = 22√ = 2–√
So the measure of ∠AMO is the inverse tangent, or arctangent, of 2–√. This turns out to be an irrational number, so this is not an example of a rational tetrahedron, as three of its angles aren’t rational. But even though it’s not what we’re looking for, this irrational tetrahedron can tell us something important about the search for rational tetrahedra.
To see this, let’s find the approximate sum of all the dihedral angles in our irrational tetrahedron. Using a calculator or a trig table, we find the approximate measure of ∠AMO to be around 54.74 degrees.
We can now sum the six dihedral angles of tetrahedron OABC: Three are right (and measure 90 degrees), and the other three are all congruent to the angle we just found. Thus, the sum of the six dihedral angles in this tetrahedron is approximately 3 × 90° + 3 × 54.74°≈ 434.22°.
Here’s where things take a turn. Let’s go back to the cube, and instead of cutting it so OA = OB = OC, imagine taking a very thin slice off the corner.
This new tetrahedron still has three 90-degree dihedral angles at OP, OC, OB, but the other dihedral angles have changed. The angle at BC now looks quite small, and the angles at PB and PC look more like the angles at OB and OC.
In fact, if you continue to take thinner and thinner slices, P will get closer to O, the dihedral angle BC will approach 0 degrees, and the dihedral angles at PB and PC will each approach 90 degrees. Notice the approximate sum of the angles:
90° + 90° + 90° + 90° + 90° + 0° = 450°.
As P gets closer to O the sum of the six dihedral angles of the tetrahedron gets closer to 450°. Which means the sum of the angles is changing! In our original tetrahedron OABC the dihedral angle measures added up to around 432°, but when we change the angles, the overall sum changes. The tetrahedron may be the 3D version of the triangle in some ways, but in one way it’s quite different: There’s no tetrahedron dihedral angle sum theorem guaranteeing that the sum of the angles is constant.
It turns out the best we can do is guarantee that the sum of the measures of the six dihedral angles of a tetrahedron lies between 360 and 540 degrees. And if you’re searching for tetrahedra with rational dihedral angles, this is a problem. You can’t just pick five rational angles and be sure the sixth is automatically rational because, unlike in a triangle, you have no idea what the sum has to be.
Even worse, you can’t be sure any six angles can even be the dihedral angles of a tetrahedron. Consider five right angles and an acute angle. The sum of these six angles is between 450 and 540 degrees, which is inside the acceptable range for a tetrahedron. But there is no tetrahedron that has these six angle measures. If five of the six dihedral angles are right, then one of the faces must have three right dihedral angles. But if this happens, those faces can’t close up and form a tetrahedron: Like parallel lines, they’ll never meet.
Faces that meet at three right dihedral angles can be part of a five-sided prism, but not a tetrahedron.
Thus, the problem of finding all possible rational tetrahedra requires much more than just finding five or six rational numbers with a certain sum. Among other things, it requires solving an equation with 105 terms inspired by a 1976 paper written by John Conway and Antonia Jones. A group of mathematicians accomplished this in 2020, and the result was a complete classification of all rational tetrahedra.
The triangle angle sum theorem is one of many reasons to admire the beauty and elegance of triangles. The lack of such a theorem for tetrahedra is a reason to appreciate their beauty and complexity, one dimension up.
Exercises
1. What is the sum of the dihedral angles in a cube?
2. What is the (approximate) sum of the six dihedral angles in a regular tetrahedron?
3. Imagine a regular tetrahedron sitting on a tabletop. As you push the top vertex down toward the bottom face, what happens to the sum of the six dihedral angles?
4. Can any four angle measures that sum to 360 degrees be the angles of a quadrilateral?
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