Since fluids flow through circular pipes, geometry is an inherent aspect of flow measurement, Traditionally, to calculate the area of a circle you need to use pi, an infinite number, and it is impossible to complete an infinite process. This leads to contradictions like Zeno’s paradoxes. Here we explain and propose a different method involving finite point and circular geometry. By requiring that the unit of measurement is specified in advance, we eliminate the need for an infinite process.
Flow and finite point geometry
Flow can be defined as the continuous and uninterrupted motion of a fluid or a pattern of objects moving uniformly along a path in a direction.
This definition captures the most common examples of flow, including river flow, the flow of a stream, flow of liquids within a pipe, and open channel flow. An open channel is the opposite of a closed pipe. In open channel flow, the flow moves by gravity, while in closed pipe flow, it is moved under pressure. Flow in partially filled pipes is considered to be open channel flow, since the liquid is not moving under pressure.
The infinity problem
Since flow seems closely connected to continuity, it is worth looking at the idea of continuity, and that brings up the concept of infinity and what I see as its inherent issues in mathematics.
The number line itself is continuous, and yet many mathematicians view the number line as being made up of discrete points. What is confusing about this analysis is that points have no area. This idea is required by the fact that it is always possible to fit another point between any two points. However, if points have no area, meaning they do not have width but are essentially dimensionless, then 1,000 points or one million points will also not have area. Mathematicians compensate for this by using the idea of infinity, arguing that even if points have no area, surely infinitely many of them will have area. But infinity multiplied by 0 is still 0 and adding infinity to dimensionless points does not yield width, length, or the number line, which is continuous.
Part of the problem with this reasoning is that a line is made up of points. But if points lie on the line instead of being a part of the line, then the line can have continuity independent of the points. In fact, one way of conceiving of a line is as the path of a point in motion. Likewise, a plane is formed by placing a line in motion.
A line is the path of a moving point
What is the relationship between points and a line? A line is the path of a moving point, as Aristotle says in De Anima 1:4,. Likewise, a plane is the path of a moving line. A point and a line then are intimately related, but not in the way Euclidean geometry describes them as being related. A line is somewhat like the trail of a meteor, except that when we use the point of a pencil to draw a line, the line is static and remains visible.
Points lie on the line not in the line
Anyone who is aware of our language will realize that we speak of points being on a line more naturally than of points being in a line. The idea that points are in a line is more a result of mathematical analysis than of an understanding of mathematical language. But what is the difference between points being on a line and points being in a line?
Someone who is sitting on a fence is not part of the fence; instead, his or her body is physically touching the fence. But no one would think that a person sitting on a fence is part of the fence. Instead, the fence is made up of steel, wood, rocks or some other material, depending on what type of fence it is. Likewise, a book lying on a table is not part of the table, although the book touches the table.
Intuitively, it makes sense to say that points lie on a line. When we draw a point on a line, typically the line is there first, and we physically mark the point on top of the line. We might us an “X” to mark the point (“X marks the spot”), with a round dot, or with a little perpendicular line. However, it is marked, it would be unusual to conceive of this “X”, round dot or perpendicular line as somehow being part of the line, while it is perfectly natural to think of the X, round dot or perpendicular line as being on the line.
If points lie on the line rather than in the line, there is no need to introduce the concept of infinity to describe how many points there are. This is because a group of points lying on a line are not continuous; instead, they are instead a group of discrete points related by all being on the same line. It is the line that is continuous, but since discrete points are not part of a continuous line, they are not part of the continuous phenomenon.
How many points lie on a line?
If we cannot consider a continuous line as an infinite set of area-less points, how should it be analyzed? We have already said that points lie on the line but are not part of the line. In this analysis, points are discrete units that sit on the line. If the number line is being considered rather than a line that consists of the distance between two points, then these points can be considered as dimensionless in the Euclidean fashion. Euclidean mathematicians will consider these as representing an infinite number of points. In a non-Euclidean analysis, these points have area. This area will vary with the unit of measurement that is specified. Pursuing this non-Euclidean conception of a point as having area or width, no matter how small, it is always possible to redefine the width or area to any level of precision desired. For example, a point can be defined as one /millionth of an inch, or one trillionth of an inch. Then there are a million points between 0 and 1, or a trillion points. These points are conceived as touching each other at the edges so there is no room for additional points once their size is determined. If a smaller or larger point is required, then the unit of measurement needs to be defined accordingly.
When boundaries matter: Defining points and lines
Sometimes, determining area by treating the boundaries of a figure as being a line with no width works well. There is often no issue about exactly where the boundary is. The boundary area between the end of one inch and the beginning of the next inch on a ruler is thought of as having no width, so that the question of where to begin and stop measuring does not arise. The lines on the ruler are lined up with the object being measured, and an inch is marked off.
Treating a boundary line as one with no width can also work in cases when the boundary line is so thin relative to what it borders that no purpose is served by treating the boundary line as having width. For example, a piece of rope that separates two tracts of land may be so thin relative to the size of the land that there is no point in specifying the boundary more precisely. Even if there is a small portion of land that lies directly on the boundary, this portion is so small that it can be ignored for the purposes of dividing the two tracts of land.
However, the boundary width may be significant when what lies on the boundary is important, or when the boundary line is large relative to the size of the marked area. For example, if gold or buried treasure lies on the boundary line between two properties, it may become important to specify which portion belongs to which property. With the center line on a highway, the line divides the two sides of highway and doesn’t belong to either side. The line is significant in size relative to the width of the road, even though it may be only several inches wide, and is an example of a boundary line with important width. Cars are not allowed to cross this line in most circumstances except to pass another car on a broken line. A doorway between two rooms provides a similar example of a boundary with width. The area within the doorway typically doesn’t belong to either room; it is there as a three-dimensional dividing line between the two rooms.
Football provides another example where physical boundaries have width. An American football field is marked off with nine 10-yard markers. Each yard line is several inches wide. If the football rests anywhere on these lines, it is “on the 10 yard line,” for example. However, at the goal line, Euclidean geometry takes over again. To score a touchdown, the player with the ball must position the ball so that it breaks the plane of the goal line before he is “down.” Here the inside edge of the goal line is treated as marking a vertical plane with no thickness that the ball must break for a goal to be scored.
In baseball, the situation is similar. Chalk lines are laid from home plate down the first and third base lines to distinguish fair from foul territory. These chalk lines are several inches wide. However, if a ball lands on the chalk line, it counts as a fair ball. It is only foul if it lands outside the chalk line. So, in this case, the chalk line is treated as an extension of fair territory. What is called the “foul pole” is really a “fair pole” since balls that hit it are considered to still be in play.
Two conceptions of points and lines
So, does it make sense to treat lines as having width? To answer this question, let us look at the function of measurement. Measuring the area or volume of an object is typically done to determine how many units of area or volume it contains. When someone is baking a cake, that person wants to know how many cups of flour he or she is putting in the cake. Likewise, quantities are important in commerce. A customer who buys a gallon of milk wants to know that she is getting one gallon, not some percentage of a gallon such as 3 ½ quarts. Two functions of measurement, then, are to specify quantities for practical matters such as recipes and to ensure that people get the advertised quantities of products.
If we treat lines as having no width, this may have no practical impact in some situations. To divide a piece of cake into two equal slices, it works to simply mark a line in the middle and physically divide the cake by cutting along the line. This act of division forces all particles of cake into one side or the other and creates two pieces of cake where formerly there was one. Of course, some crumbs may result that are particles of cake that didn’t stick to one piece or the other, but these are insignificant byproducts of cutting the cake in two pieces.
When the quantities are not being physically divided but only divided by a line, as in the border between two towns, the width of the line may be significant. In some cases, where the border is disputed, a no-man’s-land may be specified to mark an area between two provinces or countries that belongs to neither one. For example, the Korean Demilitarized Zone (DMZ) is a 160 mile long and 2.5 mile wide border between North and South Korea. It was created in 1953 as part of the armistice agreement between South and North Korea. It is a buffer zone between the two countries, it is roughly located at the 38th parallel, and is not part of either country.
In mathematical examples, theoretical problems arise in specifying the exact border or boundary of geometrical objects. (The terms “border” and “boundary” are synonyms, except that “boundary” is often used to refer to a dividing line between two areas, including countries or tracts of land. The term “border” is often used to refer to the edge of a geometric or physical object when it is not bounded by another similar area.) It is reasonable to wonder, for example, whether the area of a circle only includes the area within the circle or whether it also includes the border of the circle. This is especially true since the area of a circle is, by conventional mathematics, specified by an irrational number, so that clarifying exactly what “area of a circle” is might shed some light on our inability to specify this area with rational numbers. The same question could be asked about rectilinear figures, although the corresponding question involving irrational areas does not arise for them.
What is a line?
Whether a line has width depends on what we mean by “line.” Adolf Grunbaum comments on this issue. Speaking in the context of a discussion of Cantor’s set theory, he says:
No clear meaning can be assigned to the “division” of a line unless we specify whether we understand by “line” an entity like a sensed “continuous” chalk mark on the blackboard or the very differently continuous line of Cantor’s theory. The “continuity” of the sensed linear expanse consists essentially in its failure to exhibit visually noticeable gaps as the eye scans it from one of its extremities to the other. There are no distinct elements in the sensed “continuum” of which the seen line can be said to be a structured aggregate.” From “A Consistent Conception of the Extended Linear Continuum as an Aggregate of Unextended Elements” Philosophy of Science 19(4) (October 1952): pp. 288–306.
Does the idea that a line has no width make sense? This is Euclid’s definition, who defined a line as a “breadthless length.” (See Definition 2 in Euclid’s Book One). This idea is also consistent with Plato’s view that mathematics is about ideal, abstract objects, not about physical lines and curves. Someone who draws a rectangle and calculates the area as length ´ width (l ´ w) will not retract his statement if the lines are not completely straight or if the length and width do not form an exact 90° angle. The equation is about an abstract set of lines and relations that the drawn figure represents, not about the physically drawn rectangle.
The idea that lines do not have width was Euclid’s view, it was Plato’s view, and it is the established view of Euclidean geometry as it is taught today. Instead of arguing against this view, which most people take for granted, I am presenting an alternative conception that proceeds from a different set of assumptions. These are simply two different ways of analyzing the fundamental concepts of point, line, and area, rather than being competing geometries,. I propose a geometry in which points have area and lines have width. I believe that this geometry more closely captures how we actually conceive of points and lines in certain circumstances, as discussed above.
The idea that a line has width follows from Aristotle’s definition of a line as the path of a point in motion. The width of a line equals the diameter of the point used to draw the line. This does not mean we have abandoned Plato’s view that geometry is about idealized objects rather than physical drawings. It only means that the abstract lines represented by physically drawn lines are conceived as having width. I propose to call this wideline geometry.
Wideline geometry
Aristotle’s definition of a line as “the path of a moving point” seems preferable to Euclid’s definition of a line as a “breadthless length.” Of course, this leaves the concept of a point undefined.
It is possible to treat a line as having no width for measurement purposes. Any drawnline, no matter how thin, has some width. The width of a drawnline is similar to the duration of a unit of time. It is not possible to specify any time period that does not have duration. One hour, one minute, one second, one millisecond and one nanosecond all have some duration. Of course, sometimes it is convenient for measurement purposes to consider a unit of time such as a second as being “a point in time” with no duration. Likewise, it is useful to treat both points and lines as being dimensionless for some measurement purposes.
While drawn lines have width, Wideline geometry is not about drawn lines. Wideline geometry retains Plato’s view that drawn mathematical points and lines are about ideal objects that are represented by these drawn points and lines. The merit of Wide Line Geometry is that it coincides more closely with the way we actually treat lines in certain situations. The yard lines in an (American) football field, the chalk line on the edge of a baseball field, and the lines dividing the highway into two sides are three common examples in which we treat lines as having width. In fact, wideline geometry comes much closer to capturing the way we actually treat points and lines than does Euclidean geometry with its “breadless lengths.”
Lines, and the natural and real number lines
Some concepts are so fundamental that it is difficult to give a meaningful definition of them in more intuitive terms. Euclid’s first definition in Book One of Euclid’s Elements is “A point is that which has no part”. This definition could be criticized on the grounds that objects cast from a mold do not have parts but are not considered points. Spoons, candlesticks, statues, pots, bullets and other objects cast from molds do not have parts. While Euclid is trying to define a mathematical concept, not one that applies to physical objects, his definition still applies to these physical objects.
Euclid’s definition of a line as a “breadthless length” uses the concept of length to define what a line is, but the concepts of line and length seem very closely related. Every line has length, and Euclid makes it a matter of definition. As far as the “breadthless” part goes, this definition cannot apply to physical or drawn lines, since every drawn lines has breadth, or width. Euclid is referring to a line as an abstract mathematical concept that a physical or drawn line represents – an abstract line with no width.
Infinity and the number line
What about infinity? Aren’t there infinitely many numbers? I prefer to say, along with Descartes, that there are indefinitely many numbers. This means there is an unlimited number of numbers, but they do not exist as a completed set, as set theory proposes. This means that we will never run out of numbers, but there are not infinitely many of them.
As for irrational numbers such as the square root of 2 and π, the need for these only arises out of implicitly contradictory assumptions. The need for π only arises because we are attempting to determine the exact number of squares that fit into a circle. It is equally logical to say that there is no such number instead of calling it π and attempting to define it as a series of nonrepeating decimals that goes to infinity.
Making a measurement requires a unit of measurement and a degree of precision
When a measurement is made, a unit of measurement is either explicitly specified or understood. If I say this stick measures 4, I have not given a measurement until I specify what unit of measurement I am using. I might mean 4 centimeters, 4 inches, 4 feet, 4 yards or 4 meters. Just giving a number doesn’t state a measurement apart from a unit of measurement. Often the context makes this clear, though often the unit of measurement needs to be explicitly stated.
A degree of precision is a second requirement for a measurement. This may seem less obvious, since degrees of precision are not always specified, but a few examples should make this clear. When the weatherman gives the weather, it is almost always in whole degrees. If he predicts temperatures of 65° to 70°F, people are not going to expect him to predict to the tenth of a degree, like 65.5° to 70.3°F. It is clear just by using whole numbers that the forecasts are stated in terms of whole degrees and not in terms of tenths of a degree.
If someone gives the distance from the earth to the sun as being 93 million miles, no one is likely to demand that this number be translated into inches or feet. Furthermore, it is generally understood when this number is given that it is being given in round whole numbers, and that the number of 93,000,000 miles is most likely rounded up or down from a more precise number. Given the size of the earth, the fact that it is constantly in motion and is sometimes closer to the sun than at other times, the added “precision” of feet or inches doesn’t add any accuracy to this measurement.
Precision in time is also important in a parallel way. If I ask what time it is and you say, “It’s noon,” I am not likely to object if it is only one minute before noon. However, sometimes we need to know exactly to the minute or even to the second what the time is. This precision is often critical in making appointments and in sporting events. Even so, the time is not usually stated beyond the precision of minutes except in scientific measurements and in certain sporting events. An example is the last minutes of many professional basketball games when time is measured in tenths of a second.
There are 1,440 minutes in a day and 86,400 seconds. This doesn’t vary, except in other systems such as decimal time, but just as points have area, seconds have duration. It is easy to think that a second is a “point” in time, but a second lasts a second, a minute a minute, and an hour an hour. There is no such thing as a unit of time without duration. Otherwise, there would be infinitely many points in time, which there are not.
Sometimes an hour or even a minute can seem like an eternity. I experienced this while waiting for my Ph.D. dissertation committee to come back with their verdict after my oral exam (fortunately, it was positive, or I probably wouldn’t be writing this book). While we use the expression “point in time,” units of time are very much like points on a line. Just as units of time have duration, so points on a line have area. And the amount of area depends on the unit of measurement.
Length in flow measurement: Does a pipe circumference have width?
The concept of length is critical in flow measurement. Fluids typically travel in round pipes, and flowrate is measured by the classic equation:
Q = V ´ A
Here Q is flowrate, V is velocity and A is cross-sectional area. The area of a pipe is typically determined by the equation πr2. Here r is the radius, which is one-half of the diameter of the pipe. The inner diameter of a pipe is a straight line that runs from one side of the pipe to the other through the center of the pipe. Radius is a measurement of length and is fundamental to measuring flowrate.
It is important to distinguish between the inside diameter (ID) of a pipe and its outside diameter (OD). When measuring flow, it is the inside diameter of a pipe that is relevant to determining flowrate. While it is common in Euclidean geometry to think of the circumference of a circle as having no width, this is not how pipes are in reality. While pipes are round, for the most part, their circumference has width. The circumference of a pipe is its wall thickness. Wall thickness is important for some types of flow measurement, such as clamp-on ultrasonic flowmeters.
One common problem in determining an accurate flowrate is that buildup can occur on the inside of pipes. This impacts the accuracy of flowrate measurement since it decreases the inside pipe diameter. For example, ultrasonic flowmeter accuracy can be affected if pipe buildup occurs, since it reduces the length that the ultrasonic signal travels. Clamp-on ultrasonic meters that send an ultrasonic signal through the pipe wall can also be negatively impacted by pipe buildup. The pipe wall may already cause the signal to be attenuated, and pipe buildup can further attenuate the signal.
Circular geometry
If you studied mathematics or geometry in high school or college, you probably learned the following formula for the area of circle:
A = π x r2
Here A is the area of the circle, while r is the radius of the circle. The number π, which represents the ratio of the circumference to the diameter of the circle, is an irrational number that has never been completely specified.
What is this formula actually asking us to do? If we look at the geometry of this formula, it looks like the diagram in Figure 1.
The value r2 gives the geometric area of the square in Figure 1. The formula for the area of a circle, then tells us that π squares with sides equal to radius r fit into the area of a circle with radius r.
What, if anything, is the problem with this formula? And why do we need to have π in the formula? The reason for π is that there is no definite number of times that a square can fit inside a circle. It is often said “You can’t fit a square peg into a round hole.” This common saying reflects the insight that the area of a square cannot be used as a unit of measurement for circular area. Since there is no definite number of times that a square will fit inside a circle, the value π has to be included to create a usable formula for circular area.
The relation between circular area and square area is that they are incommensurable. What this means is that they cannot be both measured exactly using the same standard or unit of measure. Straight lines and squares work fine for squares and rectangles, but they do not allow us to provide exact values for the areas of circles.
If the areas of squares and circles cannot be measured exactly using the same unit of measurement, we have several choices. One is to continue as we are, using square area as the unit of measurement for circular area. This has the advantage of familiarity, provided we don’t mind using π. A second alternative is to use a different unit of measurement for circular area. This is the alternative I would like to suggest here.
An alternative unit of measure — the round inch
As an alternative unit of measure for circular area, I suggest the round inch. A round inch is a circle with a diameter of one inch. If we use the round inch as the unit of measure for circular area, this unit of measure looks like the drawing in Figure 2.
In Figure 2, each of the two smaller circles is equal to A/4, where A = the area of the circle. Each small circle has a diameter of one inch, and so is equal to one round inch. The area of the large circle is equal to (diameter)2. This value is equal to 22, which is equal to 4. So, the circle above has an area of four round inches.
Circular mils
A similar approach to this already exists for measuring the area of round wire. In order to avoid using decimals, the area of round wire is often measured in circular mils. The area of a circular mil is (diameter)2, so the formula is as follows:
A = (diameter)2
A mil is equal to 1/1000th of an inch (0.001 inch). A circle that has as one mil as its diameter has an area of 12 = 1. A circle that has a diameter of 4 mils has an area of 42 = 16 circular mils.
What is the relation between a round inch and circular mils? A round inch has a diameter of one inch. Since a mil is 1/1000th of an inch, a round inch has a diameter of 1000 mils. This means that a round inch has an area of 10002 circular mils, or 1,000,000 circular mils. So, any area that can be measured in round inches can be measured in circular mils, and vice versa.
Application
What is the application for circular geometry? Circular geometry can be used anywhere someone wants to measure circular area. It is true that many buildings and other structures are either square or rectangular. This is an example of geometry influencing architecture. Because most of the geometry that is taught in schools is some uncompromisingly linear, meaning that it is based on straight lines, squares, and rectangles, many of the buildings and other structures that are created using this geometry reflect its underlying linear nature. On the other hand, if we look in nature, we find a wide assortment of waves, curves, circles, and other nonlinear geometric shapes. The world is round, even though it looks flat, and many natural shapes are nonlinear as well.
One area that circular geometry has an application is in flow measurement. Pipes are round, and it is often necessary to determine the area of a pipe in order to determine volumetric flow. The formula that is usually used is the following one:
Q = A x v
In the above formula, Q is equal to volumetric flow, A is the cross-sectional area of the pipe, and v is the average velocity of the fluid. It is in providing the cross-sectional area of a pipe that the value of π is used in calculating flowrate. If this area is provided in round inches rather than square inches, flowrate can be calculated without the use of π.
Round inches can generally be substituted for square inches in geometry when calculating the area of circles. Just as circular mils are used to measure the area of wire, so round inches can be used to measure the areas of circles. Of course, just as there is no exact way to measure circular areas in terms of straight lines, there is no way to exactly measure the area of squares and rectangles using circular geometry. Just as a hammer is used for nails, and a screwdriver is used for screws, so each type of geometric structure requires its own geometry.
Note: This discussion is excerpted from New-Technology Flowmeters by Jesse Yoder (New York, 2023; CRC Press)
Blogs
Blogs about geometry