IV.4 Granular Geometry and Absolute Homogeneous Space

As we mentioned in section I.4.3, the Duality of Time Theory will have significant implications on geometry and number theory, because complex numbers are now genuinely natural, while the reals are one of their extreme approximations that are realized only when moving at the speed of light. Real numbers are the property of vacuum that is an absolute space without time, while the physical reality is evolving in space-time, thus covered by the hyperbolic granular geometry on , rather than the abstract Euclidean geometry which describes absolute space continuum, such as  or . Therefore, if we could incorporate the inner and outer components of time properly into geometry, we may be able to get rid of various persisting problems, especially those related to infinities and singularities. In order to achieve this goal, applied mathematics must obey the normal level of symmetry, which is Lorentz symmetry, because it deals with practical situations that always involve time, so it is subject to the laws of relativity. After setting the rules correctly for relativistic mathematics, we can then apply the appropriate approximations depending on the particular systems under question, just like we normally differentiate between classical and relativistic physics. This is in contrast with abstract mathematics that should be primarily understood as timeless counting , or moving at the speed of light, in all directions, in which case we can assume real and normal complex numbers  and .

For example, we cannot suppose that real numbers arereal unless we are moving at the speed of light, so whenever we want to work with real numbers we must know that we are incurring some errors that could be negligible in most situations but also could become huge and completely faulty. Abstract mathematics is the language of the divine and spiritual realms, because they are based on continuous spaces, while the physical and psychological worlds are primarily based on the granular hyperbolic geometry. We could use this abstract language in hypothetical situations, where time is not an important factor, but that should be approximated to suit the particular problem we are trying to solve. This approximation is achieved by applying the laws of Lorenz transformations on simple counting just as we apply them in relativistic physics problems. Practically, however, it may not be easy to incorporate time into the core of mathematics, but if we could do that properly, the boundaries between math and physics would vanish, and we shall find many fundamental constants and parameters in physics automatically emerging from the mathematical equations, including for example the speed of light in vacuum or in any other medium, which is the normalized value of the fractal dimension of the respective medium, as we shall discuss in section I.2.2. Also in section III.3.3 we shall see how inertial mass is generated as property of the complex-time geometry.

Mathematics started in Mesopotamia and Africa when the ancient civilizations settled around these agricultural regions, and they needed measuring and plotting the land, and also trading and calculating the taxes. Consequently, a sophisticated system of numbers and geometry was developed by the Sumerians, Babylonians and ancient Egyptians. The decimal system, using base , was used by the Egyptians, while the Babylonians used the sexagesimal system, with base . Modern mathematics is still primarily based on the decimal system, while the latter is still being used today in reporting time and measuring angles.

Common intuition indicates that numbers and counting began with the numberone, though it might have been denoted in various manners, depending on the different civilizations. Systematic counting started in Sumeria as a series of tokens represented something tangible, which was followed by the invention of writing and arithmetic. Subsequently, they transformed the number one from a unit of counting to a unit of measurement, and they used the cubit as the length equals to a man s forearm, from elbow to fingertips, plus the width of his palm. They had officially ordained sticks which they were kept in the temples, and they made many copies from these original prototypes. With this guarded and precise unit of measurement, they were able to create colossal buildings and monuments with wondrous accuracy.

After studying these ancient systems, Pythagoras introduced the mathematical concepts to the Greek, beginning with the idea of odd and even numbers, which were considered male and female, respectively. He is most famous for his Pythagorean theorem, and he was one of the world s first theoretical mathematicians, while Archimedes was the first to apply numbers on geometry. For example, he worked out how to turn the surface of a sphere into a cylinder, devising the formulas that later helped mapping the surface of the globe into a flat map. He is most famous for inventing a method of determining the volume of an object with an irregular shape. The answer came to him while he was bathing, where he was so excited that he leapt from his tub and ran naked through the streets screaming Eureka! , which is Greek for I have found it .

Nevertheless, along with this first systematic study of numbers and geometry by the Greek philosophers Pythagoras and Archimedes, similar studies also occurred at around the same time by the Indian, the Chinese, and the Mayan. India pioneered almost every field of mathematics, from the numeral system and arithmetical operations to the invention of zero and the notion of infinity. The Hindu-Arabic numerals system with its main operations, currently being used throughout the world, started developing in India in the first millennium AD, and it was transmitted to the West via Islamic mathematicians who also developed and expanded the mathematical concepts known to these civilizations. Many Greek and Arabic texts on mathematics were translated into Latin, leading to further development of mathematics in Medieval Europe, as a result of interacting with new scientific discoveries, which eventually lead to the development of infinitesimal calculus, by both Newton and Leibniz.