Topology is more general than geometry, being simply the study of CONNECTIONS (while geometry is the study of CONNECTED systems with specific SHAPE and SIZE). More specifically, TOPOLOGY STUDIES CLASSES OF SHAPES SUCH THAT ANY SHAPE IN A CLASS CAN BE TRANSFORMED INTO ANY OTHER SHAPE OF THAT CLASS WITHOUT TEARING OR RIPPING. (Thus, a circle can be topologically transformed into a square; a sphere into a pyramid; etc.)

We grapple with topology from the very beginning of our lives! Infants and small children grapple with topology in kicking off blankets; putting arms into sleeves, legs into pantlegs; in buttoning; in tying laces; in opening and closing drawers and doors; in crossing room-boundaries; etc. (Elsewhere I have topology for adults.)

A simple trick illustrates topology: taking off a vest without taking off a coat, since (topological) the vest is outside the coat -- in the sense that a paper lying on the bottom of a wastebasket is really outside the basket, not in it, since being in would require removal of a boundary. One puts an arm through one vesthole; pulls the coat through this vesthole until it is hanging on the other arm; then pulls the through that other vesthole, where it is obviously "outside".

A triangle, a square, a circle, a rectangle are all equivalent in topology! Why? Because each figure is connected within the plane in the same way. Each figure separates the plane into one inside region and one outside region.

Topologists have a special name for any figure separating the plane into one inside and one outside region: A JORDAN CURVE (named for the French mathematician, Camille Jordan (1838-1922), who first gave an enlightening discussion of this subject).

However, the figure-eight topologically differs from the square, circle, or rectangle. The figure-eight is not a Jordan curve. Why? Because the figure-eight connects differently within the plane by comparison to the Jordan curve. The figure-eight separates the plane into TWO distinct INSIDE regions and one outside region!

However, a single cut on the figure-eight transforms it into an equivalent of the Jordan curve.

NOW HEAR THIS! This idea of transforming a figure by a cut provides us with a classifier for plane figures, just as plants or animals can be classified as of this genus (generative pattern, or gnomon -- a term disussed herein).

We define the Jordan curve (triangle, circle, rectangle, etc.) as of genus zero. Why? Because the Jordan curve is the simplest of figures -- connects most simply in the plane! A triangle requires zero cuts to transform it into the simplest form; similarly, the square, etc.

On the other hand, since one cut transforms a figure-eight into a Jordan curve (genus zero structure), the topologists says that the figure-eight is of genus one.

That's correct. A figure requiring two cuts to become equivalent to a genus-zero figure is of genus two. Etc.

What? You think it's "silly" to treat the triangle, circle, square, rectangle as the same? Ah, but YOU ACT TOPOLOGICALLY IN MANY WAYS IN DAILY LIFE!

Connect two "electric" wires to poles of a battery and to leads of a small light bulb, with a switch in between. When the switch is closed, electricity flows through wires, switch and bulb, to light up the bulb. But what shape does the circuit have to be? A circle or triangle or square or rectangle or any genus-zero plane figure will achieve the same result. But a figure-eight circuit could induce capacitance effects across the touching wires! (What was connectivity-along-the-line becomes Neighbor-connectivity.) Physics students study two "Kirchoff circuitry laws", which Gustav Kirchoff (1824-87) derived by TOPOLOGICAL reasoning.

Does this remind you of a daily experience? Yes. The two-pole light switch on the wall of your residence of "living space". One flick of the switch changes the parity of a functional light bulb: if "off", its now "on", and vice versa. Two flicks of the switch returns the state of the bulb to the starting parity. Odds-and-evens again!

Kids can be taught to understand the topology of tangles -- discussed herein -- they draw on paper.

Topology is loconek or "primitive" (as we see in the daily experiences of infants and kids, and in the example of the light switch). And topology is also very hiconek. The mathematician, Edward Kasner -- whose grandson named "The Googol", discussed herein -- once said he found it easier to teach topology to kids "because they hadn't been brain-washed by geometry".

An important part of topology is the mathematical theory of knots and braids, used in understanding molecular chemistry, especially for pharmaceuticals, and in particle physics. Elsewhere, we discuss "topological conservation laws" on the frontier of physics.

TOPOLOGY is also CRITICAL IN MOLECULAR CHEMISTRY AND BIOCHEMISTRY. The great biologist, Louis Pasteur (x-y), discovered the CHIRALITY or "handedness" of molecules in discovering what made some grape mashes go bad. When light is passed through some molecular, light is ROTATED TO THE LEFT ("polarization"), in some TO THE RIGHT. Most molecular collections are a "racemic" mixture of these two types.

In the 1960s, THALIDOMIDE was given in racemic mixtures to pregnant women to treat morning sickness. The LEFT-HANDED THALIDOMIDE cured the monring sickness, but RIGHT-HANDED THALIDOMIDE caused horrible BIRTH DEFECTS, such as many children born with only stumps of arms. This was corrected. But most PHARMACEUTICAL INDUSTRIES have still not produced NONRACEMIC ("optically pure") PRODUCTS, although promising to do so. FDA (Food and Drug Administration) says it will require this in the future. "TOPOLOGY IN YOUR FUTURE!" If The Media would do their part in higlighting and pushing this matter, we might be better protected.

Yet topology is not taught! Teachers, students, citizens alike are ignorant of its presence in our daily lives and the GREAT PROMISE OF ITS RESOURCES IF WE BUT USE THEM! Using putty or playdough the TOPOLOGY part of an ATCG LABORATORY could be developed -- if only teachers knew enough or cared enough to do so.

Arithmetic is ancient. But topology (second of the MATH DNA) is the creation of the great Swiss mathematician, Leonhard Euler (1707-83), who is also the creator of the third of the MATH DNA, COMBINATORICS.