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Euclidean Geometry

Euclidean Geometry
Euclidean Geometry

The following section presupposes Set Theory.

Definition: Two distinct points \(x\) and \(z\) in a Euclidean space (simply called a space in the following) viewed as a subspace of \(\mathbb{R}^{n}\) with \(n \in {}^{\omega }\mathbb{N}^{*}\) (see Set Theory) are said to be a pair (of points). A line segment is a pair \((x, z)\) together with all inner points \(y\) in the space that are distinct from both the starting point \(x\) and the end point \(z\) and that lie between \(x\) and \(z\), satisfying \(||x – y|| + ||y – z|| = ||x – z||\) with respect to the Euclidean norm \(||\cdot||\).\(\triangle\)

Definition: Two line segments are said to intersect if they have precisely one point in common. This includes the case when the common point is only found after completing each line segment with all other inner points of that segment within \(\mathbb{R}^{n}\). A maybe one-dimensional set of points in the space with the property that each point has at least one and at most two gaplessly neighbouring points is called a line. A maximal two-dimensional subspace is named a plane.\(\triangle\)

Definition: A line segment is said to be a straight line if both its starting point and its end point lie on the boundary of the space, for the time being with the additional requirement that none of its inner points do. Two line segments are said to be parallel if one line may be obtained from the other by means of a translation or the minimum distances between each point on one line segment and the other line segment are identical. Any line segment in the space parallel to one of the straight lines defined above is also named a straight line.\(\triangle\)

Result: By defining short straight lines, arbitrarily many counterexamples can be given based on the above to Pasch’s axiom, the axiom of line completeness, as well as various other axioms and their equivalents. If a straight line uniquely defines a parallel straight line running through a given point by their shortest distance, the parallel postulate is redundant in Euclidean Geometry.

If two straight lines are only considered to be parallel when they lie in the same plane and do not intersect, then the parallel postulate does not hold: The reciprocal of the distance between the straight line and the given point may be greater than infinity or smaller than \(|{}^{\omega }\mathbb{N}|\), and then infinitely many distinct straight lines can be found that pass through the given point without intersecting the original straight line.

The Archimedean axiom must be extended to the case where a segment is marked off an infinite natural number of times without exceeding the starting point or end point of a straight line. It must be in the finite case replaced by the Archimedean theorem (see Nonstandard Analysis). Pasch’s axiom is also unnecessary, since every straight line must be fully contained in the interior of some triangle due to its maximum length, and hence so must its boundary. The hairy ball theorem does not hold:

Normal theorem: For \(n \in {}^{\omega }\mathbb{N}_{\ge 2}\), every convex set \(S\) from \({}^{\omega }\mathbb{R}^{n}\) contains a point lying on \(2n\) distinct normals through \(\partial S\) (also through a vertex), since this is possible every \(90°\) in the plane1cf. Croft, Hallard T.; Falconer, Kenneth J.; Guy, Richard K.: Unsolved Problems in Geometry; Reprint of 1st Ed.; 2013; Springer; New York; p. 14 f..\(\square\)

Toeplitz’ conjecture: Every Jordan curve admits an inscribed square.

Counterexamples: The right-angled triangle with two sides of length \(d0\) and the obtuse triangle where a vertex of at most one inscribed square is infinitesimally moved within the limits.\(\square\)

Theorem: Infinitesimally juxtaposing equichordal points within the limits leads to a Jordan domain with more than one equichordal point2cf. loc. cit. p. 9 f..\(\square\)

Fickett’s theorem: For any relative positions of two overlapping congruent rectangular \(n\)-prisms \(Q\) and \(R\) with \(n \in {}^{\omega }\mathbb{N}_{\ge 2}\) and \(m := 2n – 1\), it can be stated for the exact standard measure \(\mu\) (see Nonstandard Analysis3loc. cit. p. 25., where \(\mu\) for \(n = 2\) needs to be replaced by the Euclidean path length \(L\), that:\[\hat{m} < r := \mu(\partial Q \cap R)/\mu(\partial R \cap Q) < m.\]Proof: Since the underlying extremal problem has its maximum for rectangles with the side lengths \(s\) and \(s + 2d0\), min \(r = s/(3s – 2d0) \le r \le\) max \(r = (3s – 2d0)/s\) holds. The proof for \(n > 2\) is analogous.\(\square\)

Remark: All (three ancient) exactly unsolvable problems may be solved with arbitrary precision by the fundamental theorem of set theory via the intercept theorem and Farey sequences.4Scheid, Harald: Zahlentheorie; 1. Aufl.; 1991; Bibliographisches Institut; Mannheim, p. 62 f.

© 2010-2017 by Boris Haase



1 cf. Croft, Hallard T.; Falconer, Kenneth J.; Guy, Richard K.: Unsolved Problems in Geometry; Reprint of 1st Ed.; 2013; Springer; New York; p. 14 f.
2 cf. loc. cit. p. 9 f.
3 loc. cit. p. 25.
4 Scheid, Harald: Zahlentheorie; 1. Aufl.; 1991; Bibliographisches Institut; Mannheim, p. 62 f.