[First posted: 2.09.17]

[The following pages are excerpts from the book:

The Kinetic Foundations of Space - Conjecture from the Planck Scale.]

Book Cover

Preamble: The ideal gas law was put forward in 1834 by Emile Clapeyron to describe the relationship between the pressure, volume and temperature of a gas. It was called the ideal gas law because it treated the constituent particles in the gas as idealized points rather than as real objects with real volumes and real diameters. It was found to be sufficiently accurate as long as one worked with dilute gases where the free path was much longer than the particle diameter. As the need for a better understanding of gas behavior grew, and gas relationships were studied across a wider range of pressures and temperatures, the idealized model had to be replaced with one that better accounted for the finite, nonzero size of the gas particles. We consider whether the Pythagorean Theorem needs a similar modification when being applied to the behavior of real particles of finite, non-zero length.


Building Scales with Real Objects
When we measure length, we make reference to some object with a known dimension that can serve as our unit, then we define the total length as a multiple of this unit. We say, ‘The length of the wall is 5 feet’, or, ‘the length of a swimming pool is 50 meters‘. The units of length are ‘feet’ and ‘meters’ and the length measurement we assign to the object is the total number of unit lengths we would count if those units were placed side-by-side, adjacent to the object. But, in the real world we cannot build a ruler with abstract lines; we build our sense of length by combining objects that have spatial extent such as a ruler, and then mark unit divisions along one edge.

Accordingly, let us build a real-world object that is 5 units wide that we can use later as a short ruler. Referencing the five figures below: we start at the bottom with Figure (i) which is a line that has a length of [5u]. In Figure (ii), we form boxes of equal unit length...and, in (iii), we fill each box with a real object - in this case identical (unit) spheres. We then remove our scaffolding to leave behind the object in Fig. (iv). We can then state that this object is 'five units in length' when measured from extreme point to extreme point. As shown in per Fig. (v), that length is independent of orientation, and thus, we can manipulate it like a ruler to measure the length of some other object by placing it alongside the object.

With one real measuring device built, let us begin to establish a few general spacial relationships about lengths and the real world.

The 3-4-5 triangle
Contractors all know the following property of the ‘3-4-5 triangle’: in a right-triangle where one angle is a right angle, if one side is found to have a length of [3u] and a second side has a length [4u], then the diagonal of that triangle will have a length of [5u]. Contractors use 3-4-5 triangles to keep walls square and plumb. Underlying this easily memorable but powerful rule is the Pythagorean theorem. (On a historical note, contractors were using the Pythagorean theorem even prior to the construction of the great pyramids - and thus, prior to the birth of Pythagoras. Pythagorus generalized the relationship and then popularized it.) Contractors know this theorem not so much by its variable form: , but rather, as or as simple multiples of these numbers, such as: .

Suppose that we use the same technique that we use the same unit sphere that we used above to build the [5u] object, and build two other objects that are [4u] and [3u] long:

Then, let us then place them together to form the right-angle portion of the contractor's 3-4-5 triangle:

If we imagine this structure super-imposed on a two-dimensional X-Y grid, we would clearly state that the object we built has a width of [3u] and a height of [4u].


Now let us remove our grid and ask: what is the length of the diagonal of our triangle formed when one leg has a length of [3u] and the other a length of [4u]?

That width is shown by the question mark in yellow below. But before we answer that, let us re-assure ourselves that we have been consistent in how we label the length. Per our first diagram: length is established by the extreme end of one sphere to the extreme end of the other.

Now, assuming we have accurately positioned these two legs into a 90-degree angle, then we can say, that from the experience we have gained from six-thousand years of constructing right triangles, this diagonal should have a length of [5u]. But let us see what happens when we measure that diagonal with our [5u] ruler...

What we find, per the figure below, is that the length of the diagonal is slightly less than [5u]. This is not an illusion, nor is it due to a change in relative scales.

[Note; this may be found more dramatic to the reader if they perform this measuring task on their desk using tennis balls, billiard balls or any other group of similarly sized spheres.]

Rather, what we observe is that, when we define distance by the cumulative length of the extremities in real objects, we find that:

and that, in general, [~].

Something is not quite right here. It seems - at least in this instance - that applying a one-dimensional trait, such as distance, to an object that contains more than one dimension introduces some unforeseen changes. What we also find is that, if we do want to build a 3-4-5 triangle out of real objects, the right angle needs to be opened up a bit. This is shown below:


The angle that delivers a diagonal of length [5u] in a real object is greater than ninety degrees (greater than radians).

Distance and angle measurements defined in this way, using real objects, begin to take on very nonlinear characteristics.

The 15-20-25 Triangle
Let us examine how this distance quandary is affected by changes in scale. Below we have constructed another '3-4-5 triangle' but we have scaled our sphere's down by a factor of 5, thus, we have placed 15 unit spheres along the x-axis and 20 unit spheres along the y-axis.

The difference between the length of the diagonal of the idealized triangle shown in yellow and that of the diagonal defined by the spheres is hard to see at this scale. And, if we were to shrink our spheres by another factor of 5 the difference would be indistinguishable. But when we calculate the diagonal for the '15-20 triangle' we find it is not equal to 25 as we expect when we apply the the Pythagorean theorem - rather, we find that the diagonal is ~ 24.6.

The formula for the diagonal can be seen with reference to our original 3-4-5 figure to be given by

or, more generally with respect to changes in cordinates x and y, by:

Accordingly, we observe that the Pythagorean theorem is true for real objects only in the limit where the unit basis approaches zero and the 'construction grain' becomes indiscernible from that of an infinitely fine continuum.

A Pythagorean Gas
Let us go back now to how we started this discussion, and speak of the ideal gas law,

Let us imagine we were building our 3-4-5 triangle out of real particles spaced out to include their mean free paths like we find in a gas. Here, just to keep the image simple, we imagine their motion to be restricted to directions parallel to each respective leg of the triangle. We end up with a triangle something like that shown here:

Conceptually, we have built a local geometry that we can use to help define the gas using its own natural free path and particle geometry. We can see too, that if we are to define dynamic activity, such as the rate of momentum transfer left to right, or up and down, or along the diagonal, then - in light of our earlier discussion - the real size of the particles and the ratio of their size to the free path will play a fundamental role. Accordingly, the geometrical model we seek may not be one based on the Pythagorean theorem. While the theorem may serve as a valid first approximation when our numbers are large, it falters when we look for greater accuracy.

This is, of course, what we find when we perform experiments on gases. When we make observations on a typical gas that contains, say, one mole of the gas, we are observing activity produced by over 10^23 particles. As was shown in the formula above, when the number of partricles is large, and both a>>1 and b>>1, then the relative error introduced by use of the Pythagorean theorem is trivial in all but the most accurate observations. If we imagine reducing our scale further, by another billion-billion or so, the free path approaches the Planck-length - the grain of space-time. Any while any difference between the free path diagonal and the idealized Pythagorean diagonal would seem trivial from our perspective, it can also be argued that this small non-linear aspect of distance is what gives our world it real qualities.

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