*Geometry (n.)- the mathematical study of the properties of lines, angles, surfaces, and solids.*

This portion of the Geometry Updated Page is devoted to a foundational explanation of the historical, artistical, and literary connections between dimensionality and geometry. This section of the project serves as a basic overview for those interested in the less mathematical side of geometry. The links that are added are to benefit anyone wanting to further explore any of the items presented.

From a strictly mathematical perspective, the topic of four-dimensional geometry is a subject that still contains vast expanses of information to be explored. The study of geometry in higher dimensions is, in a sense, geometry new and updated. A basic knowlege of the history of geometry is useful when learning about higher-level geometry.

Euclid is perhaps the most famous mathematician in the realm of geometry. He created the system of rules that defines plane and solid geometry. During his time, geometry was thought to describe reality. So, in a way, Euclidean geometry was supposed to describe the real world by using a set of axioms. Even into the nineteenth century, the understanding that geometry described experience was generally agreed upon. Philosopher Immanuel Kant , author of Critique of Pure Reason , and his supporters defended this view of geometry as a model for realism.

It was not until the work of Karl Friedrich Gauss that geometry and its connection to real life experience was questioned. Regarded as one of the greatest mathematicians of all time, Gauss is considered a peer of Archimedes and Newton. Gauss helped to create a new system of geometry that did not have to exist using the axioms used by Euclid or within the two-dimensional confines of Euclidean geometry. By doing this, Gauss managed to destroy the pre-conceived notion that geometry diretly explained all experience, while, at the same time, helping to pave the way for the new of branch of mathematics known as non-Euclidean geometry. Also, among his other achievements, Gauss was one of the inventors of the electric telegraph. Apart from geometry and applied mathematics, Gauss also devoted much of his lifetime to the study of astronomical calculations including the computing of the orbit of the first asteroid, Ceres, which was discovered in 1801.

Of note, also, is that geometry, though derived from the Greeks, was not completely unknown to other ancient cultures. Though not given the specific name of geometry, both the Egyptians and the Mayans used many different geometric principles in everyday life. The Egyptians were the first civilization to discover the means with which to calculate the volume of an incomplet pyramid, and the Mayans built temples using the the step-pyramid structure which approximates the volume of a regular pyramid.

Another connection to the history of geometry invlolves the idea of different levels of education. To make the relationship between the fourth dimension and that of geometry clearer and more easily understandable, a multi-level representation is useful in providing different levels of comprehension allowing the reader to go only as far as he or she is completely comfortable with. This concept of presenting different levels was greatly influenced by Friedrich Froebel (inventor of the term kindergarten) who stressed the importance of introducing at an early age simple geometric objects so as to give a foundation with which to build on. The idea of observing simple objects helps when setting about to analyze figures that are more complex or that lie within higher dimensions.

Even in the literary tradition, there are many classical works that deal in some way or other with the fourth dimension and geometry.
Jonathan Swift's
*Gulliver's Travels*,
Madeleine L'Engle's
*A Wrinkle in Time*, and
*Republic*
all address, even if only in an indirect manner, the topic of dimensionality within the realm of basic geometry. In
Gulliver's Travels
, the dimensionality of size is covered as Gulliver travels through different lands. In *A Wrinkle in Time*, the hypercube is represented as a central projection (which resembles a cube within a cube). And finally, in the
Republic
, the

And then there is what is probably the most important piece of literature with regards to geometry and dimensionality. Edwin Abbott Abbott's book, Flatland , is a two-dimensional story that tells the tale of geometric objects, such as triangles, square, and other polygons, that live on a plane. In the book, the main character, A square, is introduced to idea of higher dimensions by a visiting sphere from the thrid dimension. The sphere attempts to show A square about higher dimensionality and ways in which A square can comprehend higher dimensional objects. The method the sphere gives to A square can be generalized so that the form of four-dimensional objects can be seen in three dimensions. And it is this analogy that is used to better understand higher dimensions for humans.

Visualizing a four-dimensional object in three-dimensional space can be better understood if the idea of visualizing a three-dimensional object in two-dimensional space is examined. In *Flatland*, to show A square that it was three-dimensional, the sphere raised and lowered its body through the plane of the Flatland surface. The size of the sphere gradually increased and then decreased, which made it possible for A square to understand the sphere to be an infinite collection of circles pieced together. This however does not mean that A square was actually able to view the sphere. When taking this from the second and third dimensions to the third and fourth dimensions, the application holds true. While we are unable to fully see a fourth dimensional object, we can visualize one using this technique. We can also infer several key components to these objects.

The role of geometry in art has also been an influential one. Many famous artists have looked to geometry for elements within their works. Piet Mondrian used straight lines throughout much of his long career. M.C. Escher also used geometric objects and perspective in many of his works. Salvador Dali's painting, "The Crucifixion," makes use of an unfolded hypercube in three-space.

Other more modern artists have used geometry as the basis for their works. Linda Henderson's book, *The Fourth Dimension and Non-Euclidean Geometry in Modern Art*, helps to visually show the reader the concepts and forms that are created through a merging of geometric principles and the fourth dimension. Other painters whose works were largely influenced by these designs are James Billmyer, whose intricate linear paintings look both random and ordered at the same time, and Tony Robbin, a more modern artist who also makes use of these techniques. Also, many artistic images have come about through the use of geometric objects and their applications within the fourth dimension such as the
Klein bottle
, the
Sierpinski gasket
, the
Koch snowflake
, and various
fractals
. On-line there is also a gallery of interactive geometry. This page of
artwork in geometry
is a useful sight for analyzing geometry in art.