Euclidean Geometry

A point is that which has no parts.

Euclid The Elements, Book One, First Definition

The beginning of all geometry: 3, 4, 6, 12, and 7

In ancient times, geometry was the predominant form of mathematics. It was to be carried out only with an unmarked ruler and a compass. This ensured that all resulting constructions and proofs were traceable without the need for measurements. The simplest form that can be drawn with a ruler is a line, while the compass is used to draw a circle.

Equilateral triangle and square originate from two equally sized circles, shifted along a straight line so that the distance between their centers equals their radius.
Equilateral Triangles: The intersection points of the two circles with the line form the vertices of two equilateral triangles (black triangles, center). These triangles are created by connecting the intersection points with the centers of the circles.
Square Construction: If the two intersection points of the circles are connected, a line is formed that serves as the diameter of a smaller circle (shaded circle). The intersection points of this smaller circle with the two larger circles, as well as the line, define the vertices of a square (gray shaded, center).
Length Ratios: The line connecting the intersection points of the large circles (gray vertical, center) corresponds to twice the sine of 60°. This line divides the distance between the centers of the circles, which corresponds to the radius, into two equal parts, corresponding to the cosine of 60°. This division occurs at a quarter and three-quarters of the diameter along the centerline.
Construction of a Hexagon and Dodecagon: A line can be drawn from the intersection points of the two circles to their intersection with the line. This creates an isosceles triangle with a vertex angle of 120°. Where this triangle intersects the inner circle, a regular hexagon (black) is formed.
By extending the construction with additional circles, whose centers are shifted by one radius length along the line (mouseover), the intersection points of the four circles can be used to draw parallel lines. This construction creates a regular dodecagon. The intersection points of the two middle circles form the center of this dodecagon (black stars).
With this method, using only two circles and possibly two additional auxiliary circles, regular 3-, 4-, 6-, and 12-sided polygons can be constructed. The necessary circles generate a total of seven relevant intersection points along the line.
3-Angle
30° 60° 120°
30° – 60° – 120°

Equilateral Triangle: The central angles are 120°, the interior angles are 60°, and the angle bisectors of the interior angles are 30°. When an equilateral triangle is rotated by 60°, a regular hexagon is formed, also known as a hexagram or Star of David (Mouseover/Tap).

4-Angle
45° 90° 90°
45° – 90°

Square: The central angles are 90°, the interior angles are also 90°, and the angle bisectors of the interior angles are 45°. When a square is rotated by 45°, a regular octagon is formed, also known as an octagram (Mouseover/Tap).

5-Angle
54° 72° 108°
54° – 72° – 108°

Regular Pentagon: The central angles are 72°, the interior angles are 108°, and the angle bisectors of the interior angles are 54°. A regular pentagon can only be constructed with knowledge of the golden ratio. All sides and diagonals are in the proportion of the golden ratio to each other (blue and orange lines: Mouseover/Tap).

Interior angles

Regular 3-, 4-, 5-, and 6-sided polygons drawn in such a way within a circle that the first angle always appears at the top center (Mouseover). The angles from the inside to the outside are: 60° (yellow), 90° (red), 108° (green), and 120° (blue).

Central angle

Regular 3-, 4-, 5-, and 6-sided polygons drawn inside a circle such that their bases are parallel. The central angles are formed from the center of the circumscribed circle to the respective bases. These angles are, from the outside to the inside: 120° (yellow), 90° (red), 72° (green), and 60° (blue).

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