An Elementary System of Axioms for Euclidean Geometry Based on Symmetry Principles

Abstract

In this article I develop an elementary system of axioms for Euclidean geometry. On one hand, the system is based on the symmetry principles which express our a priori ignorant approach to space: all places are the same to us (the homogeneity of space), all directions are the same to us (the isotropy of space) and all units of length we use to create geometric figures are the same to us (the scale invariance of space). On the other hand, through the process of algebraic simplification, this system of axioms directly provides the Weyl’s system of axioms for Euclidean geometry. The system of axioms, together with its a priori interpretation, offers new views to philosophy and pedagogy of mathematics: (1) it supports the thesis that Euclidean geometry is a priori, (2) it supports the thesis that in modern mathematics the Weyl’s system of axioms is dominant to the Euclid’s system because it reflects the a priori underlying symmetries, (3) it gives a new and promising approach to learn geometry which, through the Weyl’s system of axioms, leads from the essential geometric symmetry principles of the mathematical nature directly to modern mathematics.

Appendix

The primitive terms of the system of axioms are (i) equivalence of pairs of points (arrows): $AB\sim CD$, with the intuitive meaning that the position of the point B relative to the point A is the same as the position of the point D relative to the point C, (ii) multiplication of a pair of points (an arrow) by a real number: $\lambda , A, B \ \mapsto \ \lambda \cdot AB$, with the intuitive meaning of stretching the arrow and of iterative addition of the same arrow, and (iii) distance between points: $A, B \ \mapsto \ |AB|\in \mathbb {R}$.

• Axiom (A1) $\sim$ is an equivalence relation. (by the very idea to be in the same relative position)

• Axiom (A2) $AB\sim AC \ \rightarrow \ B=C$. (by the very idea of the relative position of points)

The definitions od inverse arrow and of adding arrows:

inverting arrow

$AB \ \mapsto \ -AB=BA$

inverting arrow

$AB, \ BC \ \mapsto \ AB+BC=AC$

Because of Axiom A2 we can extend addition of arrows:

generalized addition of arrows

$AB+CD=AB+BX$, where $BX\sim CD$, under the condition that there is such a point X.

• Axiom (A3.1) $AB\sim A^{\prime}B^{\prime} \ \rightarrow \ BA\sim B^{\prime}A^{\prime}$. (by the homogeneity principle)

• Axiom (A3.2) $AB\sim A^{\prime}B^{\prime} \ \wedge \ BC\sim B^{\prime}C^{\prime} \ \rightarrow \ AC\sim A^{\prime}C^{\prime}$. (by the homogeneity principle)

  Next axioms describe multiplication of an arrow by a real number.

• Axiom (A4) $\forall \lambda , A, B \ \exists C \ \ \lambda \cdot AB=AC$.

  (by the very idea of the multiplication as stretching arrows)

• Axiom (A5) $AB\sim CD \ \rightarrow \ \lambda AB\sim \lambda CD$.

  (by the homogeneity principle)

• Axiom (A6.1) $1\cdot AB = AB$.

  (by the very idea of the multiplication as addition of the same arrow)

• Axiom (A6.2) $\lambda \cdot AB +\mu \cdot AB = (\lambda +\mu )\cdot AB$.

  (by the homogeneity principle and by the very idea of the multiplication as iterative addition of the same arrow)

• Axiom (A6.3) $\lambda \cdot (\mu \cdot AB)=(\lambda \cdot \mu )\cdot AB$.

  (by the very idea of the multiplication as iterative addition of the same arrow)

• Axiom (A7) (the scale invariance axiom)

  If $AC = \lambda \cdot AB$ and $AC^{\prime} = \lambda \cdot AB^{\prime}$ then $CC^{\prime} \sim \lambda \cdot BB^{\prime}$.

  (by the scale invariance principle)

  Next axioms describe the distance function.

• Axiom (A8) $AB\sim CD \ \rightarrow \ |AB|=|CD|$.

  (by the homogeneity principle)

• Axiom (A9.1) $|AA|=0$.

  (by the very idea of measuring distance)

• Axiom (A9.2) $B\ne A \ \rightarrow \ |AB|>0$. (positive definiteness)

  (by the isotropy principle)

• Axiom (A9.3) $|AB|=|BA|$.

  (by the homogeneity principle)

• Axiom (A10) $|\lambda AB|=\lambda |AB|$, for $\lambda >0$.

  (by the very idea of measuring along AB)

Fig. 22

Illustration for the scale invariance axiom

The definition of a circle:

the circle with center S and radius r is $C(S,r)=\{ T: |ST| = r\}$

• Axiom (A11) If a line has two common points with a circle, points A and B, then the midpoint P of AB is the point on the line nearest to the center of the circle.

  (by the isotropy principle and an idea of continuity of space)

• Axiom (A12) If a line has exactly one common point with a circle,then the common point is the point on the line nearest to the center of the circle.

  (by the isotropy principle and an idea of continuity of space)

The definitions of perpendicularity of lines, orthogonal projection and scalar orthogonal projection of an arrow:

A line a is perpendicular to a line b, in symbols $a\bot b$, if a intersects b and there is a point S on a which is not on b such that the intersection of a and b is the point on b nearest to S.

orthogonal projection of a point S to a line p is the point on line p nearest to the point S.

The scalar orthogonal projection of the arrow AB onto the arrow CD, denoted $AB_{CD}$, is the ± length of the orthogonal projection of the arrow AB onto the line p(CD), where the sign is $+$ if the projection is in the direction of $CD, -$ otherwise.

Axiom (A13) $|AB|=|AC| \ \rightarrow \ AB_{AC}=AC_{AB}$. (Fig. 23)

Fig. 23

Illustration for the axiom A13

(by the isotropy principle)