Fig.2: Euclid (325-265B.C.E)

Euclid (325-265 B.C.E) is one of the most influential and best read mathematician of all times. His prize work “Elements” was the textbook of elementary geometry and logic up to the beginning of the twentieth century. For his work in the field, he is known as “the father of geometry” and is considered one of the great Greek mathematicians. The first six books of “Elements” cover elementary plane geometry and have served since as the basis for most beginning courses on this subject. The other books of the “Elements” treat the theory of numbers and certain problems in arithmetic and solid geometry. Euclid was also a scholar in music, art and the theory of light and optics; however, his major efforts and contributions were in mathematics. Very little is known about his life. Both the dates and places of his birth and death are known inaccurately. It is believed that he was educated at Plato's academy in Athens and stayed there until Ptolemy I invited him to teach at his newly founded university in Alexandria. There, Euclid founded the school of mathematics and remained there for the rest of his life. Personally, all accounts of Euclid describe him as a kind, fair, patient man, who quickly helped and praised the works of others.

Euclidean Geometry

Geometry is the science engaged with planar shapes such as parallel lines, triangles, rectangles, spiral as well as spatial bodies like a sphere or a cone. The word "geometry " comes from two Greek words “geo” and “metria” meaning "earth measuring." When Euclid wrote his books on geometry, he chose four undefined terms and five axioms or postulates to be the foundation of his Geometry. Every other part of geometry is built upon this foundation. Fig.3, an artwork by the artist Ibrahim Nubani (1961) living in Israel, demonstrates wonderfully different elements of the Euclidean Geometry. The fundamental quantities of the latter are “point”, “straight line”, “plane” and “space”. A point has dimensions zero, a line is one-dimensional, a plane two-dimensional, and space is three-dimensional.

Fig.3: Euclidean Geometry according to Ibrahim Nubani (1961)

In addition to Euclidean Geometry, there exist the Hyperbolic Geometry developed by the Russian N. I. Lobachevsky in 1826 and the Parabolic Geometry developed by the German Bernhard Riemann in 1854. While in Euclidean Geometry parallel lines remain at a constant distance, in hyperbolic geometry they "curve away" from each other, increasing their distance as one moves farther from the point of intersection with the common perpendicular. In parabolic geometry, the lines "curve toward" each other, and eventually intersect; therefore no parallel lines exist in elliptic geometry. The above geometries are demonstrated in Fig.4 on the basis of a modified artwork of Victor Vasarely (1906-1997), Hungarian.

Fig.4: Different geometries

The fundamental quantities of “point” and “straight line” are demonstrated below. The point is demonstrated in Fig.5 (left) by the artwork of Gino Severini (1883-1966) Italian cubist/futurist, which is composed of points. The photograph in Fig.5 (right) of Walter Wick, an American photographer, of water droplets formed at the dew point, is also a nice demonstration of a point. The straight line is demonstrated in Fig.6 by the artwork Piet Mondrian (1872-1944), Dutch Geometric constructivist.

Fig.5: Points

Fig.6: Straight lines and parallel lines

Euclid’s five axioms are demonstrated in the following by different artworks. His 1^st axiom is shown in Fig.7 by the painting of Rene Magritte (1898-1967), Belgium surrealist. The two weights simulate the two points and the bar connecting them is the straight line. Fig.8 demonstrates the 2^nd axiom by light beams, which gives the feeling that they can be extended to infinity. It is a picture photographed by Fritz W. Goro (1901-1986) a German scientific photographer.

Fig.7: Euclid’s’ 1st axiom: “Any two points can be joined by a straight line”

Fig.8: Euclid’s 2nd axiom: “Any straight line can be extended indefinitely in a straight line”

Fig.9: Euclid 3rd axiom: “Given any straight line segment, a circle can be drawn having the segment as the radius and one endpoint as center”

Euclid’s 3^rd axiom is presented by Kenneth’s Noland (1924) artwork, an American abstract expressionist where his 4^th axiom by the painting of M.C.Escher (1898-1972) a Dutch illustrator.

Fig.10: Euclid’s 4th axiom: “All right angles are congruent”

Escher’s artwork in Fig.10 is based on the impossible triangle shown on top-left. It was suggested in 1956 by Roger Penrose, but practically, already in 1934 by Oscar Reutesuard. The triangle is characterized by three right angles, so that their sum is 270°. Although it can be presented in two dimensions, it is certainly not possible to be materialized in three dimensions. Escher’s painting, “The Waterfall”, is based on this triangle, which appears in it three times. This astounding artwork demonstrates also a “perpetum mobile” because there is a continuous flow of the water in the waterfall without any pump.

Fig.11: Euclid’s 5th axiom: “Through a point outside a line can be drawn only one parallel line”

In the following basic planar and spatial configurations which complete the Euclidean’s Geometry are presented. Triangles and squares are demonstrated by the artwork of Ivan Kafka (1952) from Prague in Fig.12. Kafka used to paint leaves in different colors and afterwards spread them in forests in different geometrical shapes.

Fig.12: Triangles and squares

In Fig.13 are shown the pentagon on the left by a top view of the office of defense in Washington, and the hexagon on the right. The latter are leaves of a resam fern in the Malaysia forest.

Fig.13: Pentagon and a hexagon

Fig.14 shows the octagon. It is the artwork “Ascending of Madonna to the sky” (1526-1530) painted by Coregio (1494-1534). Fig.15 shows concentric circles. The painting by the Italian artist Andrea Mantegna (C.1431-1506) is a fresco on a ceiling in the palace Montage. The painting gives a feeling of an open ceiling through which the sky is observed. Both pictures also give the impression of depth where both artists took advantage of the effect of perspective that has been developed this time, the beginning of Renaissance.

Fig.14: The octagon

Fig.15: Circles

Figs.16 to 20 demonstrate spatial configurations. Fig.16 shows a spiral painted by Jacek Yerka (1952) a Polish surrealist where Fig.17 shows a truncated cone constructed by Ivan Kafka (1952). In Fig.18 a complete cone is shown photographed near the Eiffel Tower.

Fig.16: The spiral

Fig.17: The truncated cone

Fig.18: A cone near the Eiffel Tower

Fig.19 demonstrates different kinds of spheres. The picture on the left was painted by Rene Magritte (1898-1967). The center picture is a soap bubble photographed by the American Walter Wick where the astounding colors are due to the interference effect of light. The picture on the right was photographed in Nepal by the author.

Fig.19: Spheres

Finally, an interesting combination of a simultaneous planar and spatial combination by Victor Vasarely (1906-1997), Hungarian is presented in Fig.20. While looking at the picture patiently, once you can see a cube attached to three walls and once you see a step on which you can climb, again and again.

Fig.20: Rectangles, squares and a cube

In conclusion, the simplicity of the “Euclidean Geometry” should be emphasized, on the one hand, while being extremely usable, on the other.