Is there an explanation/theorem which explains why the triangle is so fundamental in euclidean space?

Question

I was just thinking the other day: a triangle is the simplest shape in euclidean space and so it seems obvious why all other shapes in this space could be built form it. It is kinda like the atoms of euclidean space. A square and any other polygon can be built out of triangles. The circle can be built out of infinitely many triangles.

The circle is also peculiar since it contains the most symmetry being built out of infinitely many of the most elementary shape in euclidean space, that is the triangle.

Is there a deeper theory which explains why this is so, or is it just obvious why?

Would group theory explain this at a deeper level? Is this explained by some general theorem about spaces (euclidean and non euclidean) somewhere?

If anyone has any insight as to why this is so, perhaps there is some analog in non-euclidean spaces.

Answer 1

The triangle is the simplest $2D$ polygon, i.e. shape with linear sides. This generalizes to so-called simplices in $kD$, which have $k+1$ vertices. You can indeed cover an arbitrary polygon (polyhedron) with a finite number of triangles (simplices).

But the "universality" stops there if the outline is not straight. For instance, you can very well cover a disk with infinitely many disks.

Answer 2

I can think of a few ways that triangles stand out from other shapes.

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First there is the SSS theorem:

SSS Theorem: Let $T_1,T_2$ be triangles in the Euclidean plane $\mathbb E$, with vertices listed as $A_1,B_1,C_1$ for $T_1$ and $A_2,B_2,C_2$ for $T_2$. If the lengths of corresponding sides are equal, i.e. if \begin{align*} \text{Length}\bigl(\overline{A_1 B_1}\bigr) &= \text{Length}\bigl(\overline{A_2 B_2} \bigr) \\ \text{Length}\bigl(\overline{B_1 C_1} \bigr) &= \text{Length}\bigl(\overline{A_2 B_2} \bigr) \\ \text{Length}\bigl(\overline{C_1 A_1} \bigr) &= \text{Length}\bigl(\overline{A_2 B_2} \bigr) \end{align*} then there exists a unique rigid motion $f : \mathbb E \to \mathbb E$ such that $f(A_1)=A_2$, $f(B_1)=B_2$, $f(C_1)=C_2$.

Note that the existence portion of the SSS theorem fails for $n$-sided polygons, for any $n \ge 4$. Also, for any radius $r > 0$ the existence portion of this theorem is true for circles of radius $r$, but the uniqueness fails.

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Next there is a useful generalization of the SSS theorem, in which one drops the requirement that side lengths are equal, and one replaces the strong concept of a rigid motion with the weaker concept of an affine transformation:

Theorem: For any triangles $T_1,T_2$ in the Euclidean plane $\mathbb E$, with vertices listed as $A_1,B_1,C_1$ for $T_1$ and $A_2,B_2,C_2$ for $T_2$, there exists a unique affine transformation $f : \mathbb E \to \mathbb E$ such that $f(A_1)=A_2$, $f(B_1)=B_2$, $f(C_1)=C_2$.

Again, existence fails for polygons of any number of sides $\ge 4$. And again, for circles existence is true but uniqueness fails.

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These theorems do indeed have simple group theoretic consequences, regarding the topological group of affine transformations and its closed subgroup of rigid motions.

For example, it follows that the group of rigid motions acts freely on the set of labelled triangles (i.e. triangles with vertices labelled $1,2,3$), and that the group of affine motions acts freely and transitively on the set of labelled triangles.

Also, it tells us that the right coset space of the subgroup of rigid motions inside modulo the group of affine transformations is homeomorphic to the set of triples of positive real numbers $(l,m,n)$ which satisfy the strict triangle inequalities: $$\{(l,m,n) \in \mathbb R^3 \mid l,m,n > 0 \quad l < m+n, \quad m < n+l, \quad n < l+m\} $$

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You also ask briefly about non-Euclidean spaces.

Triangles play a similar but somewhat weaker role in the hyperbolic plane on which there is constant negative curvature, and in the projective plane on which there is constant positive curvature, in that the SSS theorem holds for those geometries; although in those geometries the second theorem I stated fails.

Triangles have less significance in surfaces of nonconstant curvature, where even the SSS theorem can fail.