Euclidean geometry (and history) and construction

Euclidean geometry (also known as classical geometry, or Euclidean geometry) is based on the definitions and axioms that Euclid of Alexandria came up with in the 4th century BC in his book Foundations (Elementa).

A mathematical theory based on similar definitions and axioms is called axiomatic construction/system, which is still the formal method of building mathematical theories today. The goal of axiomatic construction is to produce a theory that is consistent (contains no contradictions), complete (allows every statement within the theory to be proved or disproved), and formal (each step is uniquely defined and independent of intuition). Thus, the claims made in these axioms are assumed to be true and are not necessary to be proven in any other way. They establish the rules of the methodology and are the pillars of further research. We first encounter the idea of axioms in the 5th-3rd centuries BC in ancient Greece, which was looking for a way to establish a solid foundation for the mathematics and geometry of the time. It was not until Euclid in the 3rd century BC, however, that he first established axiomatic construction and thus unified and consolidated the foundations of geometry at that time. The method of axiomatic construction is still used today and its solid structure led, for example, to the discovery of several important mathematical paradoxes (Russell's paradox).

The book Fundamentals (Elementa) consists of 4 books. The first book introduces the basic concepts and axioms, which then apply to the whole work and to Euclidean geometry in general. It then deals with the theory of the triangle and the parallelogram. The second book of Geometric Algebra is for the most part statements of geometric interpretations of algebraic formulas. The third book is devoted to the circle and its properties. Then the fourth deals with describing problems of inscribing figures in a circle.

Until the 19th century, Euclidean geometry was considered the only possible geometry. Only later did other non-Euclidean geometries begin to appear, such as hyperbolic or elliptic geometry today. The main difference between Euclidean and non-Euclidean geometries is that Euclidean describes a space that is flat and has zero curvature.

In the Foundations (Elementa) file in Book One, the basic concepts by which we name the various figures appearing in Euclidean geometry are introduced. These definitions include, for example, the following 7:

1. A point has no parts.
2. A line is breadthless length.
3. The ends of a line are points.
4. A straight line is a line that lies evenly with the points on itself.
5. The surface has a length and breadth only.
6. The edges of a surface are lines.
7. A plane surface is a surface that lies evenly with the straight lines on itself.

Next, the so-called axioms are introduced, from which the relations between defined objects are defined, and in fact the whole of Euclidean geometry is defined by them. These include the following 5 postulates (first tasks) and axioms (rules/policies):

1. To draw a straight line from any point to any point.
2. To produce (extend) a finite straight line continuously in a straight line.
3. To describe a circle with any center and distance (radius).
4. That all right angles are equal to one another.
5. [The parallel postulate]: That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are less than two right angles.

A Euclidean construction is then a construction in which we can only do operations that are consistent with the first three postulates/axioms. IT is important to say, that Euclides himself did not come up with these rules. They have been established later. We nonetheless call this set of rules the Euclidean geometry rules. A Euclidean construction thus defines fixed boundaries of which actions are possible, and which are not. Thus, we cannot, for example, measure distances and copy them as we proceed. According to Euclidean geometry, we have to transpose the sizes using some construction. Another important rule is, that a Euclidean construction has a definite number of steps. It is thus not possible to say, that to get to the result, we have to repeat the same step indefinitely.

Only the following tools are allowed when constructing in Euclidean construction: a ruler without a scale and a compass. This, by definition, severely limits the shapes we can accurately draw in their entirety, such as ellipses (we can only find selected points of an ellipse). Furthermore, measuring angles and their translation is not allowed. If it is necessary to draw a given angle, it must be transferred using another construction.

From today's perspective, the geometry of the ancient Greeks may seem unnecessarily constrained, as it often required long constructions that could be replaced by simple measurements. But these strict constraints had a deeper purpose: they led to the development of logical, abstract and deductive thinking. The constraints placed on Euclidean constructions can be compared to the rules of sport - clearly defined rules force "players" to seek innovative solutions to problems within the given possibilities. The use of other devices in Euclidean geometry could then be likened to breaking these rules, for example playing football with a firearm in hand. Your victory is sure and easy, but the game itself is rendered meaningless.