How to envision the universe.

We use the term come to grips with . . . And we speak of grasping a concept. If you will accept the thesis that we are the direct descendants of a hominid that mastered upright, two footed gait, and that acquired as a consequence of that the ability to use its fore legs as arms and its fore feet as hands, you might go one step further and consider that much of our brain's special abilities relate to and are likely derived from those mechanical advantages. Even mental capabilities such as abstract thinking might rest upon a neurobiological foundation laid down in the past by our ancient, savannah dwelling forebears who literally and figuratively saw their hand in front of their face.

It is said that we see only what we know. But sometimes we know only what we see--at least in the case of modeling abstract concepts. So what would be a simple model for a simple abstract like a² + b² = c²? Well, there must be close to a hundred ways to prove the Pythagorean theorem. But a simple model would utilize little square tiles, all the same size arranged in three groups to form a small, a larger, and a largest square with values 3x3=9, 4x4=16, and 5x5=25, respectively. Of course, the square of the hypotenuse of the triangle formed when the three squares are laid with tip ends touching is here 25 and is equal to the sum of the two other sides of the triangle squared. One can mentally envision the nine little squares inside the one smallest square; sixteen little sqares inside the larger square and twenty-five little squares inside the largest square. But it is graphic when done as a model of little tiles so arranged. Squared is literally squared in the model. These tiles can be set up on a table top so as to help children grasp this with their hands as well as with their minds.

Let us move from two dimensions to three while still keeping the same model and concept: Pierre de Fermat's last theorem. This is a theorem that went without a proof for centuries. It was first proposed by de Fermat in the form of a note scribbled in one margin of his copy of an ancient Greek text, Diophantus's Arithmetica. The note read in the original Latin: Cubum autem in duos cubos, aut quadrato-quadratum in duos quadrato-quadratos, et generaliter nullam in infinitum ultra quadratum potestatem in duos eiusdem nominis fas est dividere cuius rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet.(Nagell 1951, p. 252). In English: It is impossible for a cube to be the sum of two cubes, a fourth power to be the sum of two fourth powers, or in general for any number that is a power greater than the second to be the sum of two like powers. De Fermat added that he had discovered a truly marvelous demonstration of this proposition but that the margin of the page was too narrow to contain his proof. Let us envision a model for the simplest situation of this abstraction: an + bn = cn where n=3. One must see the model in three dimensions.

Envision a small, a medium, and a large cube arranged so as to make a sculpture in three dimensions. The small cube is axaxa where a=3. It is composed of twenty-seven smaller cubes as 3x3x3. The medium cube is bxbxb where b=4. It is composed of sixty-four smaller cubes as 4x4x4. And the largest cube is cxcxc where c=5. It is made up of one hundred and twenty-five smaller cubes. All the small cubes are the same size. Now attach the cubes so that their corners touch and form within a right triangle with sides three, four, and five units for the small, medium, and large cubes, respectively. This is a model for de Fermat's final theorem in its simplest form. One can see that for n=3 the x cube and the y cube do not equal the z cube in volume.

Going beyond three dimensions with the model just described requires mathematical notation and that does not lend itself to the process of envisioning. No doubt the reason for this is the inescapable fact that our species was born into three visual and palpable dimensions as were each of us as individuals and as were all of the forms of our ancestor species. We all evolved sensing but three dimensions. At least this is true of those that swam, flew, or at a minimum jumped. (If there was a flatworm in the family tree, it might have "known" but two dimensions.)

No doubt there exist more dimensions in our universe than the three we know as height, width, and depth. But light up your Cohiba in the tightly enclosed space and look for some part of that space not containing dense cigar smoke. Unless you have added a lot of Laphroaig to the mix you will not find dimensions beyond height, width, and depth. Even though our brains function better than those of our hairy, hunched back, gracile forebears on the savannah we see with our eyes and grasp with our hands but three dimensions. Perhaps fortunate for us that our mental grasp exceeds our physical grasp, at least as long as we refrain from blowing ourselves up or poisoning ourselves and our planet beyond our or its recuperative powers. But how to model in three dimensions our universe which is immense, almost infinite, really; incredibly old, and very dynamic? We know it is expanding. It is composed of energy and matter and the two are interconvertible, although not subject to destruction. We know that the laws of physics and the four forces are presumably the same throughout the entirety of the universe. We learned recently that the expansion of the universe is accelerating and that much of the universe is composed of dark matter and dark energy. We know a lot about the ultrastructure of matter, too. But what would serve us as a model of this universe of ours? Something simple enough to be easily grasped even by children?

I propose the kettle of water on the hot plate model. It helps to have good lighting for this. Set the kettle on the burner and turn on the heat. Watch closely as the first tiny bubbles form as if from nothing on the bottom of the kettle. Then the tiny bubbles let go of the bottom and begin their ascent to the top of the water. Note that the bubbles enlarge and shift their shape as they traverse the distance from the bottom to the top of the water layer. This is the model of our universe.

First observation: there is not just one bubble at a time. Likely lesson is that our universe is not universal. Probably we are one of a whole collection of such bodies but, like the bubbles that do not touch one another in their transit of the water layer, we know of no other universe. Still, the model suggests others coexist with us.

Second observation: the destiny of the universe we call home is to disappear into the void that surrounds all that is. No oscillating universe, no death in fire or ice. Just disappear into the void.

Third observation: one bubble knows only its own existence from its origin on the heated surface of the bottom of the kettle to its extinction at the surface of the water layer. And one universe knows only of its existence from its big bang origin to its eventual extinction. Even the planned array of gravity wave detectors will not disclose the existence of other universes beyond ours.

Fourth observation: energy is needed to power the model of the universe and energy is likely needed to power the universe of universes one of which we call our own.

Fifth observation: at first glance we have a couple of loose ends: matter/energy that cannot be destroyed but a whole universe of that very stuff disappearing into the void at the end of its run. Ditto for all the other sister universes. And a whole lot of energy powering up the kettle on the other end of the system.

Sixth observation: the loose ends can be connected, at least a concept of that can be grasped. Even with our feeble, three dimension accustomed minds.

Envision for the fun of it one more scene: dark clouds of a thunderstorm, a bolt of lightning striking a lone tree on the savannah, a dozen hairy, hunched, and upright forms watching awestruck as one of their number reaches into the burning pyre to extract a flaming branch. If our bold forebear could grasp (manually and mentally) the concept of fire, we can create a model so as to envision our universe.