The reality of a mean free volume does not imply that its geometry always that of a sphere. As the word ‘topology’ is defined, a cube, an ellipsoid, a cylinder, a cone, a bean shaped object, or any closed surface without hole(s) are all topologically equivalent to the sphere. However, their applications require either minimum or maximum surface area. An example of minimum surface area is the shape of rain drop or soap bubble. Maximum surface area is found in red blood cells allowing maximum absorption of oxygen in the blood stream. If this maximum is compromised then the onset of a blood disease called sickle cell anemia emerged. On the other hand, a white blood cell can change its shape between maximum and minimum as needed to protect the body from foreign invasion of bacterium, virus, or inorganic material. Therefore, the red and the white blood cells, the hairy bacterium, the leggy virus all are topologically equivalent to the sphere. These would include the coverings of basketball, football, and golf ball. The first is designed for maximum handling and control, the second is designed for directional pointing, and the third has tiny indentations designed for maximum travel distance. Nevertheless, all these are topologically equivalent to the sphere.

On the other hand, sphere, donut, and pretzel are not topologically equivalent as differentiated by their characteristic property of having hole(s). Some donuts have one hole and pretzel has several holes. By this property, a teacup with one handle and a donut with one hole are topologically equivalent to each other. In other words, they have the same topology. This characteristic property is called the genus. It is defined as the maximum number of times a surface can be cut along simple closed curves without the surface separating into disconnected parts. It is the same as the number of handles on the surface. Since each handle creates a hole, no matter how large or how small, genus is also defined as the number of hole of a given topology. The Möbius topology is unique since its genus can be negative as well as positive defined as directional genus whose one dimensional transformation becomes a Hopf link.

Negative and positive genus can be understood by its mathematical connection to the vector space approach to understanding geometric shapes in 3-space. It is a defining concept of oriented volume as a physical quantity given by the outer product of three vectors A, B, and C denoted by [ABC]. If this represents a right-handed system then its negative represents a left-handed system. Odd permutations of A, B, and C change the sign of the oriented volume while even permutations preserve the sign. There are 3 axioms for oriented volume: (1) skew-symmetry, (2) linearity, and (3) zero volume. Sign changes define skew-symmetry, linearity is indicated by the distributed product [AB(C+D)]=[ABC]+[ABD] and [AB(��C)]=��[ABC] where �� is a scalar quantity. [ABC]=0 when, and only when, A, B, and C are linearly dependent. Consequently, the algebra of Hadamard matrices represents a mathematical theory for the distribution and association of oriented volumes in multidimensional spacetime and each element of a given Hadamard matrix, which can either be symmetrical or non-symmetrical, represents a particular unit of oriented volume of either positive or negative unity. Nonetheless, each symmetric Hadamard matrix of equal number of positive and negative unity represents either of two basic configurations of spacetime charges: H-plus and H-minus. Either one can be found embedded in an infinite order matrix of whole numbers minus unity called a sieve of Diophantus as the inverse of any two by two submatrix of a base matrix plus multiples of the particular Hadamard matrix. More about oriented volume can be found in the book by Melvin Hausner’s ‘A Vector Space Approach to Geometry’, Dover edition, 1998.