More on Circular Geometry
A Circular Geometry of Flow
In a recent article,1 I argued that a new geometry of flow is needed. In this article, I expand on this argument. I begin by describing the flawed foundations of Euclidean and Cartesian geometry. I then present the axioms of a new Circular Geometry. Finally, I discuss the implications of this geometry for flow measurement.2
The Flawed Foundations of Euclidean-Cartesian Geometry
Traditional geometry rests on two main pillars: the Euclidean axioms that create the conceptual underpinning of geometry, and the Cartesian coordinate system that provides an x and y axis (and z axis, in 3-dimensional geometry) in terms of which points can be located. Unfortunately, both systems have fatal flaws.
One flaw in Euclidean geometry consists in the Euclidean conception of the point, and the relation between points and lines. In Euclidean geometry, a line is made up of infinitely many points, each of which has zero area. No matter how many times zeros are added together, however, a positive value never results. And adding infinity into the equation doesn't change anything, since zero times infinity is still zero. While many attempts have been made to explain away this anomaly, it still remains unexplained and unexplainable.
There is also a flaw in the Cartesian coordinate system. While the straight-line framework of the Cartesian coordinate system works well for analyzing the areas of squares and rectangles, it is less successful as a frame of reference for curved and circular areas. Finding the area of a circle requires the use of pi, which is the ratio of the circumference of a circle to its diameter. Since the formula for circular area, pi * (r * r), involves the area of a square, (r * r), this formula actually involves calculating how many squares will fit into a circle. Since a square peg does not fit into a round hole, it should not be surprising that no definite number of squares will fit inside a circle. In fact, the number is pi, a nonrepeating, irrational number that mathematicians to this day cannot fully define.
The need for pi arises because square or rectangular area and circular area are incommensurable. This means they cannot both be analyzed from the same perspective, or point of view. Instead, it is necessary to adopt a frame of reference that is appropriate to circular area to adequately describe and analyze circular area.
It is time to eliminate the flawed elements of Euclidean-Cartesian geometry. The belief that a line is composed of infinitely many points with no area is simply incoherent, and should be abandoned. Likewise, the idea that it is possible to accurately represent circular or even curved area in terms of the straight-line frame of reference supplied by the Cartesian coordinate system should also be discarded.
A New Circular Geometry
Instead of analyzing circular area in terms of square inches, I propose to analyze it in terms of round inches. If we start with the primitive of the round inch in place of the square inch, it is possible to represent circular area in terms of how many round inches fit into a circle. It is then possible to eliminate pi from the equation, and substitute in its place a different formula that has no need of pi. I am proposing to call this new geometry Circular Geometry.
Like Euclidís geometry, Circular Geometry can be formulated as a series of axioms and definitions. These axioms are formulated in the following section. Within Circular Geometry, terms such as 'point' and 'line' receive a different interpretation from the traditional one. To account for this, in this discussion, terms such as 'Point' that have a corresponding meaning in traditional geometry are capitalized to reflect their use in the sense specified by Circular Geometry. 3
The Axioms of Circular Geometry
1. A Point is the smallest allowable unit of measurement within a system of measurement.
2. Every Point has area.
3. A Point is a circular figure that is considered to be indivisible for the measurement being made.
4. A Line is the path of a Point in motion.
5. Every Line has width. The width of a Line is the diameter of the Point being used for a particular measurement.
6. Every measurement involving Points and Lines is relative to a system of measurement in which the reference of the terms 'Point' and 'Line' is fixed for the purposes of that measurement. The reference of 'Point' determines the degree of precision used in the measurement.
7. In the Coordinate System of Circular Geometry, there is an X axis consisting of unit Circles laid end to end in an east to west direction, each with a value of one round inch. Likewise, there is a Y axis consisting of unit Circles laid end to end in a north to south direction, each with a value of one round inch. The Point of intersection of these Circles creates the integers 0, 1, 2, 3, etc.4
8. To find the area of a Circle, which is the number of round inches in the Circle, use the formula 4 * r * r, where r is the radius of a Circle. Alternatively, use d * d, where d is the diameter of the Circle.
9. A Point lies on the Line, not in the Line. Finitely many Points lie on any given Line.
10. A Circle is a continuous closed geometric figure that is the path generated by rotating a Point around a fixed Point of the same size until it intersects its starting position.5
11. Every Circle has an inside Diameter (ID) and an outside Diameter (OD). The inside Diameter being and ends at the inside edge of the Circle. The outside Diameter begins and ends at the outside edge of the Circle. The inside area is calculated by taking d to be the Inside Diameter in the formula d * d. The outside area includes the Circle boundary, and is calculated by taking d as the outside Diameter in the formula d * d.
12. A Plane is the path generated by moving a Line through Space. Just as a Line has width, a Plane has a depth equal to the width of the Line.
The Coordinate System of Circular Geometry is shown in Figure 1. Rather than amplifying on Circular Geometry in this article, I will discuss the implications of Circular Geometry for flow measurement. Does this new Circular Geometry have any implications for flow measurement? I believe that the answer is a resounding "Yes!" Here are three areas in which Circular Geometry has the possibility of improving flow calculations.
The Fundamental Unit of Flow Measurement
The first area has to do with the fundamental units of flow measurement. The fluid dynamics section of a college physics text describes two approaches to measuring flow6:
"One way of describing the motion of a fluid is to divide it into infinitesimal volume elements, which we may call fluid particles, and to follow the motion of each of these particles. This is a formidable task. We could give coordinates x, y, z to each such fluid particle and then specify these as functions of the time t and the initial position of the particle x0, y0, and z0. This procedure is a direct generalization of the concepts of particle mechanics developed by Joseph Louis Lagrange."
A second approach was taken by Leonhard Euler, which Halliday and Resnick describe as follows:
"In it we give up the attempt to specify the history of each fluid particle and instead specify the density and velocity of the fluid at each point in space at each instant of time. This is the method we shall follow here. We describe the motion of the fluid by specifying the density r(x, y, z, t) and the velocity v(x, y, z, t) at the point (x, y, z) at the time t. We focus attention on what is happening at a particular point in space at a particular time, rather than on what is happening to a particular fluid particle. Any quantity used in describing the state of the fluid, for example, the pressure p, will have a definite value at each point in space and at each instant of time. Although this description of fluid motion focuses attention on a point in space rather than on a fluid particle, we cannot avoid following the fluid particles themselves, at least for short time intervals dt. For it is the particles, after all, and not the space points, to which the laws of mechanics apply."
In Circular Geometry, flow is not measured in infinitesimal volume elements, but in small "flow units" (Points) whose size would vary (or could vary) with the fluid being measured. These "flow units" would be defined in terms of Circular area, rather than square area. It is like the particle approach, except that the volume elements are not infinitesimal, but instead are finite and definable. Then the amount or quantity of flow is given in terms of how many of the finite flow elements travel past a given location in time. A parallel intuition would be to measure flow in drops e.g., to say "This fluid is flowing at 1,596 drops per minute." A conversion could also be created from drops (or Points) to gallons or liters.
More specifically, the "flow units" I am proposing are the Points of the Circular Geometry laid out in the above axioms. These Points could either be defined in terms of volume or of mass. Fluid flow through a pipe is then described in terms of how many of these "flow units" or Points pass a given location in a period of time.
Application to the Flow Equation
A second way to apply this geometry is as follows. In the flow equation:
Q = V * A
where flow volume = velocity times area, area can now be measured in round inches instead of in square inches. This eliminates pi from the equation, and does away with a nagging source of inaccuracy and uncertainty in flow calculations.
Inside Diameter and Outside Diameter
A third application is that, in this geometry, a Circle has an inside diameter and an outside diameter. This inside/outside diameter corresponds naturally to pipes, which also have an inside and outside diameter. When calculating flow, it is the inside diameter that is used.
I began by describing the flawed foundations of Euclidean and Cartesian geometry. I then presented the axioms of a new Circular Geometry. Finally, I discussed the implications of this geometry for flow measurement. Circular geometry has implications for flow measurement in terms of the fundamental unit of flow measurement, the flow equation Q = V * A, and in terms of measuring the inside and outside diameters of pipes.
In conclusion, it is time for a fresh look at the assumptions of Euclidean-Cartesian geometry. And while much work remains to be done, I believe that Circular Geometry will prove to be a viable alternative to Euclidean and Cartesian geometry. Change is often good, and in this case, change is needed. Not only can Circular Geometry put geometry on a more sound footing, it can also improve our methods of measuring fluid flow.7
I would be grateful for any comments on these ideas. Feel free to contact me at the following address:
27 Water Street
Wakefield, MA 01880
Our website: http://www.flowresearch.com
1. I&CS, November 1997 (Radnor, PA: Chilton Company, pp. 25-31)
2. While I first conceived the outlines of Circular Geometry in 1979 when I was a philosophy graduate student at the University of Massachusetts, I was not able to formulate it in axiomatic form until very recently. My ability to do so now mainly arose out of extensive discussions on the online Geometry Forum, whose Internet address is:
Discussions with John Conway, Kirby Urner, K. K. G. Yau, Floor Lamoen, Candice Hebden, Helena Verrill, DGoncz, Clifford Nelson, and Heidi Burgiel were very helpful in forcing me to reduce my arguments to their fundamental form.
3. The need for a convention of this type was pointed out to me by Dr. John Conway of Princeton Universityís mathematics department.
4. Comments from Dr. John Conway also were especially helpful in formulating Axiom 7. Conway challenged me to consider a purely Euclidean-Cartesian interpretation of Circular Geometry, which is still another possible geometry. In this Euclidean-Cartesian circular geometry, circular area is still analyzed and described in terms of round inches instead of square inches, but the terms ëpoint,í ëline,í ëcircle,í ëplane,í and ëspaceí have their traditional Euclidean-Cartesian interpretation.
5. I am indebted to David Clayton for a comment that caused me to modify Axiom 10.
6. Fundamentals of Physics, 1981, by David Halliday and Robert Resnick (New York: John Wiley & Sons, p. 277)
7. Discussions with my students in over 20 philosophy classes at the University of Massachusetts Lowell and Lafayette College from 1986-1995 were very helpful in developing my ideas on Circular Geometry. The discussion of flow measurement is based on twelve years experience in writing about process control and flowmeters, and on hands-on experience in my flowmeter laboratory. David Spitzer of Nepera (Harriman, New York) was very helpful in explaining flow concepts to me in his ISA (International Society of Measurement and Control) course on flowmeters and flow measurement. In addition, discussions with the following people over the past nine years of market research have helped me better understand flow measurement: John Schnake (Honeywell), Terry Barck (Rosemount), Dick Verville (Honeywell), George Mattingly (NIST), Jim Gray (Krohne), Stan Rud (Rosemount), Tom OíBanion (MicroMotion), John Garnett (Dieterich Standard), Sam Herb (Moore Products), and Bob Harvey (Honeywell).
NOTE: See Figure 1 on next page.
More on Circular Geometry
27 Water Street
Wakefield, MA 01880