Zero and Infinity In mathematics, we see these two numbers appear almost always as the end points of some limiting processes. We see zero in the definition of a derivative in calculus. We see infinity in the math of infinite series. In this discussion, we will try to see these extreme numbers in physics. These will be used to give some insights into the following statements: (1) The mass is zero, (2) the mass is infinite, (3) the time is zero, and (4) the time is infinite. The mass is zero. Since mass is equivalent to energy, the statement is same as “the energy is zero.” Zero energy does have a meaning in quantum mechanics. It is the energy of the vacuum equals half a quantum (Planck’s constant over 4 pi). This is associated to a point particle at the absolute temperature of zero. Because of Heisenberg’s uncertainty principle, this energy can never be exactly zero. This gives meaning to the quantum vacuum as a complex structure. This structure was verified by experiments to detect the Casimir effect. This structure of the quantum must necessarily give meaning to an infinitesimal volume. In mathematics, we can define volume in any given dimension of space. And if the geometries are that of a sphere or a cube, there exist formulae to calculate their volumes. It can be shown that the volume of 1-sphere is equal to the volume of a 1-cube. The different in volume becomes worse and worse at higher dimensions. This volume is static. In order to make the volume dynamic, it must be associated with a vector, in this case, an infinitesimal vector. This is called a local infinitesimal motion (LIM). For volumes to have any physical meaning, it must be a closed volume. The closure of one-dimensional volumes forms closed one-dim loops. And there are two distinct types. The LIMs are all conserved and there is infinite number of them. Here again we have reached infinity by not going as far as the next LIM. In mathematics, a line segment or a loop (closed line segment) contains in itself infinite number of points. We can take away the end points of a line segment, and still there remain an infinite number of points. It is useless to keep removing end point, the number of points will never be less than infinity. But in the loop, the LIMs have started from a begin point and have passed infinity and back to the same begin point without really knowing it, just in the blink of an eye. And it continue to cycle through these infinite points forever and ever holding on to the same bearing of travel. This is the principle of a directional invariance. If we attempt to visualize this volume in two-dimension, we get the limiting problem resulting in a Sierpinski carpet, which area is zero, and while the total perimeter of the loop is infinite. In three dimensions, we get the Menger sponge, whose volume is zero, while its surrounding area is infinite. Only in one dimension does the volume of the loop make any little sense. And it is in this one-dim that we can justifiably defined eight invariant properties for the principle of directional invariance. The time is zero. This is just to give a linear definition for time. Taking the line segment again, zero time is surely at one end. The other end might be when time equals infinity. But in a closed loop, time just keeps flowing, from time zero, passes infinity and back to zero, on and on. Since in principle, there are two distinct types of closed loops from a 360 degrees twist of a Moebius strip, there are two directions of time. These two directions can only make any sense if viewed in one dimension. If we attempt to view time in three dimensions, we get the senseless image of Escher’s Waterfall. The validity of most of Escher’s images is true only if we live in the one dimension. For the one dimensional Waterfall, it is logical to have two directions of flow. In three dimensions, the water (time) has to flow uphill fighting against the forces of gravity making it only natural for water (time) to flow downhill. We have discussed the concepts of 1-dim volume and the possibility of two directions of time. By so doing, without even realizing what we have done, we have actually separated the space-time structure of special and general relativity and came up with a bonus of an extra time’s direction than can most benefit the description of antimatter, which was not part of relativity but is a big equal participant in quantum field theories.