Fig.11d demonstrates how the identical bodies shown below are reflected in different sizes due to the perspective effect of the surroundings.

Fig.12 demonstrates the Gynus and Horsehead nebula. A nebula is a huge cloud of gas, which due to gravitational concentration, form the different bodies in the universe. The illusion of the 3D effect is generated by the difference in the size of the bodies due to their different distances in the nebula. Douglas Finkbeiner photographed the Gynus nebula.

Fig.12: 3D space of Gynus Nebula (left) and Horsehead Nebula (right)

Fig.13 is an additional demonstration of 3D spaces. Fig.13a is a cat in a crate painted by the American S.S.David (1847-1898). Fig.13b was painted by the American Andy Warhol (1928-1987) while Angelo Michele Colonna and Agostino Mitelli painted Fig.13c.

Fig.13: 3D spaces

Fig.14, by Pablo Picasso (1881-1973), a Spanish painter and sculptor, demonstrates a unique style in art that has been developed by him and by Georges Braque (1882-1963), a French painter. This style is called Analytical Cubism and lasted from 1909 through 1911. Cubism is based on the simultaneous presentation of multiple views where in the picture below front and profile of a portrait appear simultaneously. In other words, the above combination demonstrates a 3D image.

Fig.14: Picasso’s portraits demonstrating 3D spaces due to simultaneous presentation of front and profile

Figs.15-19 demonstrate Escher’s astoundingly different kinds of multidimensional spaces. Fig.15 entitled “Another world II“ describes the interior of a cube-shaped building. Through the openings, three spaces are seen, each a 3D one. So we have in this artwork nine dimensions. From the topmost pair one looks down, almost vertically, onto the ground. The middle two are at eye-level and show the horizon, while through the bottom pair one looks straight up to the stars.

Fig.16, “Relativity”, demonstrates again three spaces or worlds. Here we have three forces of gravity working perpendicularly to one another. Three earth-planes cut across each other at right angles, and human beings are living on each of them without interaction. They also do not have knowledge of each other’s existence.

In Fig.17 a “Double planetoid” is demonstrated. This little planet inhabited by humans has the shape of a regular tetrahedron and is encircled by a spherical atmosphere. Two of the four triangular surfaces, with which this body is faced, are visible. The edges that separate them, divide the picture into two. All the vertical lines: the walls, houses, trees and people, point in the direction of the core of the body-its center of gravity-and all the horizontal surfaces, gardens roads, stretches of water in pools and canals, are parts of the spherical crust.

Fig.18, “Gravitation”, is a stellar dodecahedron demonstrating a multidimensional space, the subject of this article. This interesting space may be regarded as being constructed in various different ways. Inwardly it consists of a regular twelve-faced body (a dodecahedron), each face of which is a regular pentagon. And upon each of these faces there is superimposed a regular, five-sided pyramid. In these worlds lives a tailless monster with a long neck and four legs.

Fig.15: Three worlds

Fig.16: Three worlds

Fig.17: “Double planetoid”

Fig.18: A multidimensional space

According to Einstein’s special relativity, space and time are interrelated and their magnitude depends on the relative velocity between an object and its observer.  In other words, every event and measurement is viewed differently depending on the observer's velocity where the only quantity that remains constant is the speed of light C in vacuum.  It is also the maximum speed in nature.  In other words Einstein’s Universe is a four-dimension space, namely, X-Y-Z and speed V.  In Fig.19,”Depth”, space is divided up cubically.  Each fish is found at the intersection of three lines of fish, all of which cross each other at right angles demonstrating the dimensions X-Y-Z.  The fourth dimension is reflected in Escher’s artwork by the impression that the fish are moving at some speed V.

Fig.19: Einstein’s 4D universe

References:

• Ron Miller and William K.Hartman, “The Grand Tour”, Workman Publishing, New York (1993)
• Marc Grossman, O.D. and Rachel Cooper, “Magic Eye”, Andrews and McMeel (1995)
• M.C. Escher, “The Graphic Work”, Benedict Taschen (1990)
• Jacob D.Bekenstein, “Information in the Holographic Universe”, Scientific American, Inc.,59-65 (2003)