| prev |
CHAPTER VI |
next |
In this chapter I present a consistent model for each of the two positions. The traditional number line is developed and examined for the purpose of contrast with the less familiar view -- atomism.
Flour is finely grained stuff which has been being divided for millennia. Have you ever tried to measure a cup of flour? One cup, more or less, is likely to differ from another by minute amounts. Water is another divisible quantity. While the grains of flour can be made visible by a sufficiently strong magnifying glass, the "grains" of water cannot. True, we have obtained electron micrographs which seem to show the individual atoms of some of the heavier metals. But the same still cannot be said of water.
We are accustomed to dividing "stuff". To assist in dividing stuff, we have devised units of measure. We have invented arbitrary units of measure and conversions between them. For example, to convert from liters to gallons we multiply the number of liters by 0.2641720524.(1) The process of dividing materials, in conjunction with measuring how much (quantity), has developed into the use of numbers for measuring and dividing.
The fifth or sixth century equivalent of a Certified Public Accountant kept track of inventories. One may reasonably presume that counting filled containers along with adding tallies was sufficient for inventory purposes. But, when the auditors came along, subtraction was necessary to determine how much there "ought to have been". Tallying the last inventory, adding the records of the amounts received, and subtracting the records of the amounts drawn from stores would yield how much a current inventory should find.
With the advent of subtraction it is only a matter of time before negative numbers are needed. When the plans for issues from stock, say in planning for a battle or a trip, are compared with the inventory and found wanting, this wanting can be quantified by negative numbers. "We need 100 sacks of flour more", the planner might say. The answer to "How many will we have after the trip?" becomes "minus 100".
Inventing multiplication merely needed an extremely rich person or a ruler with too many inventories to tally directly. Adding the same size tally many "times" yields how many "times" we have that tally. It isn't easy to tell the ruler with 45 granaries and a tally of 150 sacks of flour in each granary how many sacks of flour he has. We soon learned how to multiply 150 "times" 45.
Division follows quickly with the need to allocate stocks evenly to a number of storage locations, or to the legions of soldiers. How much do we have? How many battalions of legionnaires must we divide this among? Finding out how many supplies to take away and give to each battalion is a long and cumbersome process when trial and error subtraction is used. The bigger the bureaucracy, the more the need there is for division as a tool. With a notation for numbers in place, it's just a matter of time before someone devises a better way; division is that way. But when the answer doesn't come out even, the result is a fraction of a whole. Since simple fractions (dividing in two, etc.) were not unfamiliar, that division sometimes yields them makes it less strange.(2)
It falls to the philosophers to analyze these relations and develop a theory of pure numbers -- numbers which are viewed apart from that which they might be applied to in measuring or counting. The '3' in '3 bags of flour' is seen as something existing apart from the bags of flour.
There is a natural progression in this development. First there are tally strokes. Tally strokes are measured by numbers which count the tally strokes. Addition of numbers becomes a shorthand for tallying a lot of tallies. This process generates the natural numbers. Whenever any two natural numbers are added, the result is a natural number. The natural numbers are said to be closed under addition. If something is added to something else and a result is obtained, then one ought to be able to subtract something from the result and obtain something else. This process works well enough for some pairs of natural numbers, but not for other pairs. Three less one gives two, but one less three does not give any known natural number. It gives the number which, when added to two, gives zero. We extend the natural numbers to include this zero as well as the other strange numbers, which we call the negative integers.
When we include these negative integers, zero, and the positive integers and thereby obtain simply the integers, we find that the integers are closed under both addition and subtraction.
We found that adding a natural number to itself many times is tedious and invented or discovered a shorthand -- multiplying that natural number by the number of times we took it as an addend. We found that the natural numbers are closed under multiplication. We also found that the integers are closed under multiplication. One strange effect was noted -- the product of two negative integers is a positive integer.
Naturally, if one thing is multiplied by something else and a product is obtained, one might well ask how the one thing might be obtained from the product and something else. Since a product is obtained by adding one thing many times, the one thing could likewise be subtracted many times -- distributed among the many somethings or divided among them. In such a case the one thing is called the divisor and the other factor, which when multiplied by the divisor gives the product, is called the quotient.
A similar problem to the one which arose in subtraction arises. For some pairs of numbers, taking one as a product and the other as a divisor, dividing the divisor into the product yields a known integer. For example, six divided among (by) two gives three. But for some of these pairs of numbers, taking the first as a divisor and the other as a product, dividing the divisor into the product does not yield a known integer. For example, two divided among six does not yield any known integer, positive or negative. Each of the six only gets a "fraction" of a whole. Initially, fractions, as such new numbers are called, are added to our growing list of types of numbers. Combining fractions with the integers gives what we call rational numbers. We notice one more anomaly. Zero cannot be divided into other numbers. It seems an ad-hoc solution, but we just forbid division by zero. It doesn't work.
I've given a hypothetical account of how numbers might logically have been developed. For the purpose of this work, rational numbers are all we need. Let's look at the properties we pretty much take for granted of the rational numbers. Whenever any two rational numbers are added, multiplied, subtracted, or divided (excluding the forbidden division by zero), the result is another rational number. The rational numbers are closed under the four arithmetic operations -- addition, subtraction, multiplication, and division.
The relation "less than" (or "greater than") has been taken largely for granted. Higher counts are greater than lower counts. Using this relation, it is found that any two number must satisfy one of three relations (the trichotomy).
When a number is greater than a first number and less than a third number it is said to be between the other two. It is easy to show that given any two distinct rational numbers, it is possible to find another between the other two. For example, consider the rational numbers 1/3 and 1/2. 1/3 is the same as 4/12; 1/2 is the same as 6/12. Clearly 5/12 is between 4/12 and 6/12. Since 4/12 is 1/3 and 6/12 is 1/2, 5/12 is between 1/3 and 1/2. As a general procedure, one may add 1/2 the difference between the numbers to the smaller number. This will always yield a number less than the higher number and larger than the lower number, or between the two given numbers.
The property of having another element between any two given members is called denseness. A set of numbers is dense if and only if between any two numbers in the set there is another number in the set.
The natural development of numbers for counting and measuring originally had the numbers intimately associated with the stuff being counted or measured. At these practical levels numbers were never separated from the things they were a measure of. It was only with the invention of "pure" numbers, numbers taken apart from the things they were traditionally used to count or measure, that the properties of numbers could be separated from the properties of the things they were used to measure. And the properties of numbers drives the questions about the properties of the stuff they are used to measure.
When rational numbers are used in measuring the quantity of stuff, it is presumed that the stuff is as divisible as are the numbers. More in question is the so-called "extension" of stuff rather than its matter. The rational number system we use to measure "how much" (stuff) with has the property of denseness. We can continue dividing between numbers as long as we like. The question that arises naturally is "can the stuff we associate the numbers intimately with be similarly divided?" Even talk of (pure) extension dissociates the "distance across" some stuff from its matter. When we remove the matter from consideration we talk of pure "extension".
We are accustomed to measuring extension relative to directions. Two (non-parallel) directions are required to measure area; three are required for volume. We can certainly conceptualize the (empty) space as something apart from the system of numbers we use to measure it. But when we ask "how much" in regard to such extension, we talk about its "measure" -- the numbers we use to describe how much. Talk of divisibility also asks after the "stuff", including the empty space, we use the numbers to measure.
We are posed with a complex question. We have a consistent model for measuring divisibility, one that exhibits denseness, and can therefore support infinite divisibility. We also have only conceptualization as a way of holding onto the concept of the extension of empty space (or of matter). But our conceptualization is amenable to using the model provided by numbers. We conceptualize the difference between two distinct points in a visualized blow-up where we can picture ourselves walking (part way) from one point to the other. So we naturally argue, by analogy, that "pure extension" is similarly divisible. One point of the present work is that, sufficiently informed, the analogy is not so obvious.
In the following section I present a much less familiar model, one which presumes that the model provided by the rational numbers does not apply, Once that model has been presented it will no longer be such an "intuitively obvious" conclusion that the "stuff" of empty space is best modeled by the rational (or real) number system. The fifth postulate of Euclidean geometry was once thought so intuitively obvious it was taken to be a "self evident" truth, yet we now know of self-consistent geometries based on "less intuitive" statements of the postulate.
In this section I build and examine a consistent model which is based upon the premiss of an indivisible minimum extension. Computer graphics screen displays implement this model.
But although most geometricians from the time of Euclid have in fact worked with the principle of infinite divisibility, mathematicians do not refuse to consider the possibility of a geometry of finite divisibility.(3)
A model of extension using atomic magnitudes, while not consistent with infinite divisibility, is not inconsistent by itself. An analogy with geometry will serve to illuminate my view. For millennia people tried to prove the insight that parallel lines never meet -- the so-called Euclidean or fifth postulate. More recently the fifth postulate was shown to be independent of the others. This independence allows constructing a variety of non-Euclidean geometries. They are each internally consistent but generally not compatible with each other. Each geometry depends upon the form chosen for the fifth postulate. In a similar manner people have argued for the intuition that extension is infinitely divisible. As in the history of geometry, in which various arguments were thought to prove the Euclidean form of the fifth postulate, various arguments have been advanced as refuting atomism. Two modern views belie these historical alleged refutations of atomism.
One view is that provided by the invention of the microscope. Microscopically granular substances appear continuous at macroscopic levels. Modern particle physics has found a hierarchy of successively smaller particles cumulating in quarks and leptons which are indivisible -- so far. Matter, strictly speaking, is not extension, but the extension of matter appears to be quantized (atomic), although at smaller levels than the namesake. Extension per se can be quantized with a discrete metric.
The other view derives from the advent of computer graphics. Computer display screens exhibit de facto atomic extension. The smallest portion of a display is called a pixel (picture element). IBM PC Monochrome Graphics Adapter (MGA) and Color Graphics Adapter (CGA) displays have 640 horizontal by 200 vertical pixels. Enhanced Graphics Adapter (EGA) displays have 640 by 350 pixels. Vector Graphics Adapters (VGA) displays have 640 by 480 pixels. Drawing a line on one of these graphics displays requires turning on successive or contiguous pixels. A minimum length line consists of two adjacent pixels (points). Except on very high resolution displays, lines not aligned with the axes appear as small step functions. Even on very high resolution displays, lines appear as step functions when viewed through a magnifier.
Now that computer displays have become commonplace, they may be used as an example for illustrating discrete metrics. By their very nature they implement discrete metrics, and they serve as a good example to illuminate a certain "weirdness" inherent in discrete metrics.
Let a display be specified as composed of an NxM array of pixels. Drawing a square on such a field requires using the same number of pixels across as up and down. Drawing the diagonal, however, uses only one pixel for each of the vertical and horizontal positions. There are exactly the same number of pixels in the diagonal as there are in both the horizontal and the vertical sides. Figure 8 is an example of a 7x7 square with one diagonal drawn on such an array.
The usual metric with the Pythagorean criteria of preserving distance with any rotation cannot be presumed to hold. The distance between adjacent pixels is 1 minimum unit, no matter what the direction. The shape of the pixel affects the "size" of the distance according to some external criterion. Within the system, there is no way to discern that the diagonal distance differs from horizontal or vertical distances. I shall discuss these differing "sizes", but I shall have to deal with a few preliminaries first.
In addition to squares, triangles and hexagons also form plane-filling patterns using pixels which exhibit some kind of radial symmetry. Hexagons seem likely choices as they approximate circles or "dots". There are "more" directions which have the same "distance" (0, 60, 120, etc.). Hexagons allow 6 directions of symmetry, but unfortunately, hexagons cannot "slide" past one another, while triangles and squares can. Also, hexagons are made up of smaller triangles. Triangles allow three directions of symmetry, while squares allow two.
When it comes to three-space, cubes form space-filling solids. Equilateral pyramids do also. Cubes allow three directions of "sliding", while pyramids allow four.
But specifying a position in two-space requires only two coordinates; similarly, specifying a position in three-space requires only three coordinates. Consequently, the simplex pattern (triangle, pyramid) has a redundancy in its directions of movement resulting in a loss of 1 degree of freedom. Parsimony is achieved by requiring orthogonality in each additional dimension. Squares and cubes are therefore the logical choice for conceiving atomic pixels. However, using squares for illustrations prevents distinguishing individual pixels. To allow distinguishing individual pixels, I shall use circles for illustrations.
We are not accustomed to thinking of extension in terms of atomic distances. In fact, we are so accustomed to thinking of extension as being continuous that we have difficulty even conceiving of it as being atomic. One branch of mathematics which covers theories of distance is metric space theory. A theory of distance which has integral units of distance is called a discrete metric. The distance from a point to itself is always zero in a metric space, discrete metrics included. Under an atomic theory of extension, there is no zero unit of extension, although there is a zero unit of distance. The measure of distance between two points is different from the extension or length of the line that includes those two points. The distance from a point to itself is always zero, even though the extension of the point itself is not. The distance from a point to its nearest neighbor is one unit of extension, but the length of the line including the two points is two units of extension.
The length of a line composed of only two points is actually a minimum of two units of extension. The continuous length of such a line can be larger than two units of extension. The ratio of the length of such a line to one with the minimum extension may not be 1:1. One can conceptualize these differences in "distance" using the concept "aspect ratio".
The ratio of the horizontal and vertical distances is usually not one to one on computer displays. For example, my EGA display on a 13" monitor has 640 pixels wide by 350 pixels high displayed in an area which is 10 inches wide by 7.5 inches high. The ratio of the horizontal height to the vertical width of a picture or screen is called the 'aspect ratio'.(4) On my display, which is 10 inches wide by 7.5 inches high, the aspect ratio is 10/7.5, which is 1.33 or 4:3.
Drawing lines on such a display is also affected by the shape of pixels themselves. In the case of my EGA display, there are 640 pixels in 10 inches horizontally and 350 pixels in 7.5 inches vertically. The horizontal size of a pixel is 10"/640. The vertical size of a pixel is 7.5"/350. The ratio of these is 0.729 -- 0.729 = (10/640)/(7.5/350) or 35:48. The screen itself is wider than it is tall, but each pixel is taller than it is wide. These facts must be taken into consideration when drawing pictures on such displays. If one presumes that the aspect ratio of a pixel is 1:1 when one draws pictures on such a display, the resulting pictures will appear distorted.
When a picture is stored or recorded using one aspect ratio and reproduced using another aspect ratio the resulting view will also appear distorted. This effect can be illustrated by a familiar technology in film. Wide screen, or cinemascope(5), motion pictures do not use wider films to store and project the wider picture.(6) Cinemascope pictures use standard 35 millimeter films. How can the wider picture be stored on the film? A special lens, called an anamorphic lens, is used which distorts the image on the film by shrinking it in the horizontal direction.
The aspect ratio of the cinemascope frame is 23.8:18.67 = 1.275 instead of the normal 1.38. Since the anamorphic system operates with a ratio of 1:2 the effective screen aspect ratio will be (23.8/18.67)2 = 2.55.(7)
The image of a square will appear on the film as a rectangle which is narrower by half than it is tall, and the image of certain double wide rectangles will appear on the film as squares. The image is stored on the film with a "perspective ratio"(8) which is not 1:1. Because the horizontal compression is twice the vertical compression, the perspective ratio on film is 1:2. When a cinemascope motion picture is projected a special anamorphic projection lens is used to widen the image back to its original proportions.(9)
A picture recorded using a cinemascope lens will be stored with a perspective ratio which is 1:2. If that picture is projected using a standard lens it will be projected presuming a perspective ratio of 1:1. A cinemascope picture projected using a standard lens will appear squeezed together and too tall. I have seen Cinemascope pictures appear this way on television while the credits are running. Conversely, a standard picture projected with a cinemascope lens will appear stretched out and too short. A standard film is stored with and projected with a 1:1 perspective ratio. A cinemascope film is stored with and projected with a perspective ratio which is 1:2. Each picture will appear normal when it is projected using the same perspective ratio with which it was stored. But when either picture is projected using a different perspective ratio from that with which it was stored, it will appear distorted.
Anyone who has tried to draw a low resolution picture on a computer screen using a character, say the asterisk, can see the effect immediately. A square number of asterisks, say 6x6, hardly looks like a square. And when one finally gets something that looks reasonable on the display screen, it looks different when it is printed.
The aspect ratio of most display screens for characters is about .42.(10) To make a square using six lines of text one would need 6 divided by .42 or 14.4 characters on each line. But what looks like a square on the screen prints out as too wide. The aspect ratio for printed text in a 10 characters per inch by 6 lines per inch mode is .6.(11) The printed square using six lines of text would require a width of only 10 characters. In figure 9, the left array is numerically square. On the screen the middle one looks square, and on the printed page the right one looks square.
Measuring distances using a metric based on the atomic premiss requires that any "distance" be in terms of multiples of the minimum distance, the distance between adjacent points. The concept of perspective ratio can be adapted to give us a quantitative measure of the difference, in non-atomic terms, between the scales used for different directions. Pixel aspect ratios show that one perspective can illuminate (or obfuscate) the other, as when the square on the display screen doesn't print square. But one perspective (atomic or continuous) cannot be used to evaluate the other lest a contradiction be introduced in the overall system. Like the wave particle duality of matter, transformation equations must be rigorously (and religiously) used when switching perspectives.
Under the atomic presumption there are adjacent points, and the minimum length of a line segment consisting of two adjacent points is two minimum units. We can only visualize these points as "dots" of a fixed size and indeterminate shape. We would like to presume that these dots can be thought of as small disks and will do so for illustrative purposes, but in consideration of the foregoing discussion of the aspect ratio of pixels, we must be ready to cast this assumption aside. Some argue that this assumption might entail a contrary presumption -- that angle is infinitely divisible, a question dealt with elsewhere.
A line between two points may have two sides. Figure 10 is an illustration on a 7x9 array of pixels. The line from point A to point B passes directly through point O but passes immediately to the right of points represented by a left semicircle and to the left of points represented by a right semicircle.
A closed plane figure would have to include either the left semicircle points or the right semicircle points depending upon on which side of the line the figure was. Both figures would include points A, O, and B. Obviously, the degree of overlap depends upon the orientation of the line as well as its length. As a practical matter, computer implementations of line drawing functions presume a "preferred direction". Turning on the rightmost one of a horizontal pair of pixels and the lower one of a vertical pair is only one of 4 possible implementation strategies and is shown in figure 11.
Before we can intelligently discuss overlapping plane figures, we must examine intersecting line segments. Two lines intersect at a point, and in atomic or discrete metrics a point has a finite size or a minimum length. For the following discussion, lengths (and areas) are given in terms of the minimum unit of size. If the length of two intersecting line segments are A and B, then the length of the combined segment is A + B -1. One point is shared by both line segments, and its size must be subtracted. If A and B were merely added, as we are accustomed to doing with continuous distances, the overlapping point would be counted twice; its length must be subtracted. Figure 12 is an illustration.
Of course, there are also line segments which cross each other but which do not share an actual intersection point. Figure 13 shows such a case. Line segment AB has a slope of -2; segment CD has a slope of 1. The positioning of these lines is such that there is no point that both line segments pass through. If the only points the line actually passed through were selected in an implementation, segment AB would appear as a sequence of dots.
To "thicken" the line and make it more visible, the points immediately to the left of or to the right of the line must be selected. Figure 14 identifies the points which would be selected were the line thickened "to the left". In both figure figure 13 and figure 14, line segment AB "crosses" line segment CD but does not pass through a point on segment CD. In such cases the extension of the two line segments taken together does add up to the sum of the individual extensions. The geometric interpretation of this is satisfied for finite geometries. In figure 15, the line is thickened to the right. This option implements the strategy mentioned illustrated in figure 11 above. In this case one of the points used to thicken line segment AB is one of the points on line segment CD. There is a shared point of intersection in this case, but it is not located where it would be located were the lines continuous. In such cases the extension of the two line segments adds up to the sum of the individual extensions less the extension of a single point.
When it comes to drawing plane figures, we must rely on our experience with continuous metrics, but we must be prepared for "weird" (unintuitive) differences.
Let us examine some "minimum sized" triangles with atomic magnitudes. Clearly the smallest has two sides each of length 2 (points). Since the two side lines intersect at a point, one point is shared by both lines. Rotation aside, there are only two ways to draw such a figure.Figure 16 is a degenerate triangle -- a line segment consisting of two intersecting collinear line segments. Figure 17, on the other hand, is a recognizable triangle. Notice that the diagonal has a length of 2 points, hence is the same "size" as the other two sides. However, the perspective ratio between the diagonal and the horizontal (or vertical) is not 1:1. In fact, it is 1.414:1. Perspective ratio relates ratios of atomic distances (number of pixels) to continuous (infinitely divisible) distances along different directions or dimensions. We must beware that we do not evaluate atomic figures from the perspective of presuming infinite divisibility. To do so would be to beg the question or, worse yet, introduce a contradiction, from which anything follows.
All this seems a little strange when held up against our conventional view, which is based upon continuous lines. However, lest we fall into the same trap the ancients did, it behooves us to develop a little familiarity with the atomic perspective.
Drawing a triangle in a discrete metric space so that the length of its sides in continuous distances is comparable to the number of pixels on the line requires some ingenuity. There is a way which can make maximum use of our familiarity with continuous metrics. Locate the vertex points on the centers of pixels as shown in figure 18. Draw continuous (divisible) lines connecting the vertexes. Examine the center point of each pixel interior to the continuous triangle or overlapping it. If the center point of the pixel falls on or inside the triangle, then count this pixel as part of the area of the triangle. The illustrations help guide our understanding, but we must develop a formula for the area of a triangle which can be compared to the familiar continuous formula.
The area of an atomic triangle is not simply ½B·H. We can compute the area by devising a mapping from the continuous plane to the atomic plane. Here's how. Draw a right triangle with the vertices centered on atomic points so that the base B and height H cover the requisite number of atomic points as in figure 19. Next fill in the points on the base line and the height line as in figure 20. Then draw the diagonal line connecting the two vertices, and fill in the points which are on or interior to that line as in figure 21.
We can compute accurately the number of points to fill in without actually drawing the line by noting the relation between distance along the base and the proportions of the triangle. We are, in effect, computing the height of each similar triangle which has an integer number of points along the line of the base. In figure 22, the smaller triangle and the larger triangle are similar. This similarity may be expressed in a precise proportion. Side h is to side b as side H is to side B -- h:b::H:B. The corresponding mathematical formula, h/b=H/B allows us to compute side h; h=(H/B)·b. Even though the extension of the sides is the number of points and is given by B and H, the length in continuous distances is actually B-1 and H-1. (The starting point is not counted in measuring distances, but must be counted in measuring the atomic extension of the sides.) If we count out to the Ith point along the line of the base, we can compute how many points are under the (continuous) diagonal by using the proportion in continuous distances. Because of the relationship between length and extension, the actual length of the base of the smaller continuous triangle is 1 less than the number of points. The length of the base of the triangle with I points is I-1. The length, in continuous distance, of the vertical line under the diagonal is computed using the appropriate proportion of the height. When the appropriate values are inserted the atomic version of the formula becomes ((H-1)/(B-1))·(I-1) or (I-1)·(H-1)/(B-1). If we take the integer part of this we will have the length in atomic distance units of the height of the triangle with I points along the base. But the extension of that line is one point more than its continuous length. The expression for the extension of this line is INT((I-1)(H-1)/(B-1))+1 -- which is just the total number of points in the vertical column of points at the Ith point along the base. Adding those extensions for each point along the base gives us the total area of the triangle in atomic points -- the SUM from I = 1 to B of INT((I-1)(H-1)/(B-1)+1).
The area of an atomic triangle with base B and height H is
INT((I-1)·((H-1)/(B-1))+1); it is not simply ½BH. Figure 23 is a table of the areas of discrete triangles up to 10x10. The values are computed using the above formula. It can be shown that this formula reduces to the familiar
½BH as the size of the individual points approaches zero.(12)
I have developed a consistent mathematical formula, but it is not always clear exactly what the drawings look like when compared to corresponding continuous figures. Figure 24 shows various small discrete (right) triangles. Tabulated with each one is its size and the lengths of the opposite and adjacent sides (A and B).
Armed with some familiarity with intersecting (and non-intersecting) lines and simple triangles, we are now in a position to examine overlapping plane figures. Consider the square with a diagonal in Figure 25.
Drawing a diagonal across a square usually divides the square into two triangles. It is no different in atomic metrics. But, because there is a minimum extension in atomic metrics, the line that forms the diagonal is itself extended.
This line is a part of both triangles; its extension is therefore included in the extension of each of the two triangles. Consequently, the area of the two triangles, which includes the area of the line twice, is larger than the area of the square. The area of triangle ABC is 28 units of extension. The area of triangle BCD is also 28 units of extension. But the sum of the areas of triangles ABC and BCD -- 28 + 28 = 56 -- is larger than the area of the square -- 7 x 7 = 49. The area of their common line, BC, is 7. The area of square ABDC is the sum of the areas of triangles ABC and BCD less the area of the common line -- 28 + 28 - 7 = 49.
This property must be taken into consideration very carefully whenever traditional geometric demonstrations are attempted. The areas of adjacent figures do not just add up to the area of the figure they comprise. Any such demonstrations must be reexamined in the light of the atomic perspective, if a corresponding atomic demonstration is to be made. The results of such demonstrations are often different from those in continuous geometries. One such demonstration involves the traditional (non-atomic) "proof" of the Pythagorean theorem; this theorem has figured into alleged proofs of infinite divisibility.