22 One square and an odd number of triangles

Definition 22.1 valutaion on \(\mathbb {R}\)

Definition 22.2 Three-coloring of plane

TODO

Definition 22.3 Rainbow triangle

TODO

Lemma 22.4

For any blue point \(p_0 = (x_b, y_b)\), green point \((x_g, y_g)\), and red point \((x_r, y_r)\), the \(v\)-value of the determinant

\[ \det \begin{bmatrix} x_b & y_b & 1 \\ x_g & y_g & 1 \\ x_r & y_r & 1 \end{bmatrix} \]

is at least \(1\).

Proof ▶

TODO

Corollary 22.5

Any line of the plane receives at most two different colors. The area of a rainbow triangle cannot be \(0\), and it cannot be \(\frac{1}{n}\) for odd \(n\).

Proof ▶

Follow from 22.4

Lemma 22.6

Every dissection of the unit square \(S = [0, 1]^2\) into finitely many triangles contains an odd number of rainbow triangles, and thus at least one.

Proof ▶

TODO

Theorem 22.7 Monsky’s theorem

It is not possible to dissect a square into an odd number of triangles of equal algebra area.

Proof ▶

TODO

Appendix: Extending valuations

Lemma 22.8

A proper subring \(R\subset K\) is a valuation ring with respect to some valuation \(v\) into some ordered group \(G\) if and only if \(K = R \cup R^{-1}\).

Proof ▶

TODO

Theorem 22.9

The field of real numbers \(\mathbb {R}\) has a non-Archimedean valuation to an ordered abelian group

\[ v: \mathbb {R} \to \{ 0\} \cup G \]

such that \(v(\frac{1}{2}) {\gt} 1\).

Proof ▶

TODO