Aristotle's wheel paradox #4

Wikipedia has a page for Aristotle's Wheel Paradox. I edited it substantially during September-December 2018. The page was very poorly written before then. I added two original solutions, the second and third below. The page has been edited many times since then, but only in minor ways. On August 18, 2021 somebody else edited the page, greatly reducing in size two images I had put on the page and putting them into frames on the right side of the page. Wikipedia allows an editor to preview how his or her edits will affect the page's appearance on the device he or she is using. However, the page's appearance on my smart phone -- and likely most or all other smart phones -- is quite different due to its small screen size. For example, the frames do not appear. Their content still shows, but the article's flow and appearance are worse.  

Most Wikipedia pages can be edited by millions of people whenever they get the urge. So I am hereby preserving part of the article as it existed before August 18 in this blog, where presumably nobody but me can modify it. I will not so preserve the section titled History of the paradox, which I did not edit. The following are the introduction and the Analysis and solutions section, which were written near 100% by me. 

Introduction

Aristotle's wheel paradox is a paradox or problem appearing in the Ancient Greek work Mechanica traditionally attributed to Aristotle. A wheel can be depicted in two dimensions using two circles. The larger circle is tangent to a horizontal surface (e.g. a road) that it can roll on. The smaller circle has the same center and is rigidly affixed to the larger one. The smaller circle could depict the bead of a tire, a rim the tire is mounted on, an axle, etc. Assume the larger circle rolls without slipping (or skidding) for a full revolution. The distances moved by both circles are the same length, as depicted by the blue and red dashed lines and the distance between the two black vertical lines. The distance for the larger circle equals its circumference, but the distance for the smaller circle is longer than its circumference: a paradox or problem.

The paradox is not limited to a wheel. Other things depicted in two dimensions show the same behavior. A roll of tape does. A typical round bottle or jar rolled on its side does; the smaller circle depicting the mouth or neck of the bottle or jar.

There are a few things that would be depicted with the brown horizontal line in the image tangent to the smaller circle rather than the larger one. Examples are a typical train wheel, which has a flange, or a barbell straddling a bench. In this case the the distances moved by both circles with one revolution would equal the circumference of smaller inner circle. A similar but not identical analysis would apply.

Analysis and solutions

First solution

The paradox is that the smaller inner circle moves 2πR, the circumference of the larger outer circle with radius R, rather than its own circumference. If the inner circle were rolled separately, it would move 2πr, its own circumference with radius r. The inner circle is not separate but rigidly connected to the larger. So 2πr is a red herring. The inner circle's center is relevant, its radius is relevant, but its circumference is not.

Second solution

This solution considers the transition from starting to ending positions. Let Pb be a point on the bigger circle and Ps be a point on the smaller circle, both on the same radius. For convenience, assume they are both directly below the center, analogous to both hands of a clock pointing towards six. Pb travels a cycloid path and Ps a curtate cycloid path as they roll together one revolution.

While each travels 2πR horizontally from start to end, Ps's cycloid path is shorter and more efficient than Pb's. Pb travels farther above and farther below the center's path – the only straight one – than does Ps. The image below shows the circles before and after rolling one revolution. It shows the motions of the center, Pb, and Ps, with Pb and Ps starting and ending at the top of their circles. The green dash line is the center's motion. The blue dash curve shows Pb's motion. The red dash curve shows Ps's motion. Ps's path is clearly shorter than Pb's. The closer Ps is to the center, the shorter, more direct, and closer to the green line its path is.

If Pb and Ps were anywhere else on their respective circles, the curved paths would be the same length. Summarizing, the smaller circle moves horizontally 2πR because any point on the smaller circle travels a shorter, more direct path than any point on the larger circle.

Third solution

This solution only compares the starting and ending positions. The larger circle and the smaller circle have the same center. If said center is moved, both circles move the same distance, which is a necessary property of translation and equals 2πR in the experiment. Also, every other point on both circles has the same position relative to the center before and after rolling one revolution (or any other integer count of revolutions). For a wheel with multiple concentric inner circles, each circle's translation movement is identical because all have the identical center. This further proves the circumference of any inner circle is entirely irrelevant (when the outer, larger circle is the one that rolls on a surface).

Aristotle's wheel paradox #1  Feb 18, 2018

Aristotle's wheel paradox #2  Feb 22, 2018

Aristotle's Wheel Paradox #3 Nov 10, 2018

Teen: "Math isn't real" Revisited

Three months ago I commented on a teenager who made a video saying “Math isn’t real.” Link. The video went viral. She also said the math she learned in school is real. She was obviously confused and naive. She asked three related questions: 

1. How did people come up with algebraic formulas?
2. How did they know what they were looking for?
3. How did they know they were correct?

I gave answers to them then. However, yesterday something else occurred to me about the second question. Why did she assume one has to know what one is looking for in order to find it?

Maybe it is partly her age and experience. Being a teenager, searching on the Internet became a common activity before she was born. She likely does it a lot. For that activity, one does need to be aware of what one is looking for in order to find it. Sometimes when one is looking for something quite specific, one has to already be aware of exactly what one is looking for. Else the search engine returns can overwhelm what one is looking for. What you are looking for will be buried deep in the stack and thus hard to find. Whereas using optimal search terms will put what you are looking for at the top of the stack or close thereto.

I learned that when I did a search about myself and something specific. What I was looking for was very local news. Hence, it was reported in only one not widely read publication. Searching only for my name on Google, there were 1500 hits. That's with a very unique name (first and last together). What I was looking for was well below the middle of the stack. Searching for my name and two more terms, what I was looking for was on top of the stack. Imagine the more general search had millions of hits.  

This may seem like an insignificant point. However, it is more significant from a philosophical perspective. It is an aspect of Empiricism. Experience shapes and limits knowledge.

Teen: "Math isn't real"

A teenager, Gracie Cunningham, posted a video on TikTok of herself talking about math. In this video she says she doesn’t think math is real, then next says the math she learned in school is real. She was obviously confused.

She received a lot of flack about it and made a second video.

A mathematician came to her defense and called the teen’s questions “profound.”

I wouldn’t call her questions profound or stupid, but do call them naive.

Three related questions the teen asked are: 1. How did people come up with algebraic formulas? 

2. How did they know what they were looking for?  3. How did they know they were correct? 

Maybe Gracie could answer these herself if somebody asked her leading questions the way Socrates did the slave boy about geometry in Plato’s Meno.  I won’t do that here, but what follows could guide the leading questions.

I’ll take the second question first. Why does she believe somebody has to be looking for something in order to find it? Suppose Gracie is walking on a sidewalk and she sees a $5 bill that somebody else (no longer present) dropped. Did she need to be looking for a $5 bill in order to find it? 

Suppose she walks further and finds a $1 bill that was also dropped by somebody else (no longer present). She may have been more on the lookout for money this time, but was she looking for $5 + $1 = $6? 

The roots of algebra can be traced to the ancient Babylonians. It was formalized later, especially by the Greek mathematician Diophantus and Persian mathematician al-Khwarizmi. Both men have been regarded as "the father of algebra" (link).  

I suspect some people before Diophantus and al-Khwarizmi were implicitly doing algebra. Maybe they used a question mark, not X. Suppose a farmer acquired some more land and wonders what amount of wheat seeds he needs to plant. Suppose he  knows he used four 50-lb sacks per acre last year and he has 80 acres now. His neighbor just gave him 300 pounds of seeds to pay him for something. So how many more 50-lb sacks of seed does he need this year for planting? 50*X = 200*80 – 300. So X = (200*80 – 300)/50 = 26. That is an answer to question #1. After he plants all the seeds and finds he had the correct amount, he has an answer to question #3.

An Australian math or stats lecturer says Gracie’s questions are “insightful” (link). 

There are some good reasons for questioning the reality of math. For example, imaginary numbers aren’t real numbers. Maybe Gracie will make a video about imaginary numbers some day. i² = −1 looked pretty unreal to me when I first saw it. I later learned it is also a really powerful concept.

Little math problem

For pi day here is a little mathematical proof problem for math geeks.

For every natural number n, 20 + 21 + ...+ 2n = 2n+1 − 1.

It's true for n = 1 to 5 as follows:

n=1: 20 + 21 = 3 = 22 − 1

n=2: 20 + 21 + 22 = 7 = 23 − 1

n=3: 20 + 21 + 22 + 23 = 15 = 24 − 1

n=4: 20 + 21 + 22 + 23 + 24 = 31 = 25 − 1

n=5: 20 + 21 + 22 + 23 + 24 + 25 = 63 = 26 − 1

The general formula for all n is proven by mathematical induction as follows and shown here.

Base case: Set n = 0. Then 2⁰ = 1 = 2¹ − 1.

Induction step: Show that if the equation holds for any particular n, which the above does, 
it also holds for n+1.
Let n be any natural number and 2⁰ + 2¹ + ... + 2ⁿ = 2ⁿ⁺¹ − 1 is true.

Then 2⁰ + 2¹ + ...+ 2ⁿ + 2ⁿ⁺¹ = ( 2⁰ + 2¹ + ... + 2ⁿ ) + 2ⁿ⁺¹

                                               = ( 2ⁿ⁺¹ - 1) + 2ⁿ⁺¹

                                               = 2 * 2ⁿ⁺¹ - 1

                                               = 2ⁿ⁺² - 1. QED

There is another proof as well. I will post it this weekend to give readers some time to try to find it on their own.

+++++++++++

March 16. The other proof follows. 2⁰ + 2¹ + ...+ 2ⁿ is a sum of a geometric progression. Using long division, the quotient of (xn+1 − 1)÷(x - 1) is xn + xn-1 + xn-2 +... 1 = xn + xn-1 + xn-2 +...+ x0.
Reverse the order of the sum and let x = 2:
20 + 21 + ... + 2n = (2n+1 − 1)÷(2 - 1) = 2n+1 − 1.

Aristotle's wheel paradox #2

A rolling wheel is not a simple motion. While rolling is rotation plus translation, only the latter really matters for the paradox. Still, it is easy to be lured by the complexity. For example:

1. A point on the perimeter of a wheel travels a cycloid path. A more inner point travels a curtate cycloid path. In the paradox the center's path is a straight horizontal line.

2. A point's velocity in the direction the center moves is faster than the center's velocity during the top half of a rotation. A point's velocity in the direction the center moves is slower than the center's velocity during the bottom half of a rotation.

3. Pure rolling occurs when the circular object travels one circumference along the ground for every for every full rotation it makes. Slipping occurs when rotation is faster than pure rolling. A paradigm case is a car wheel stuck in snow. Skidding occurs when rotation is slower than pure rolling. A paradigm case is a car wheel that skids on ice after the driver brakes hard.

Slipping and skidding so described affect the two circles in the same way. For example, if the part of the tire in contact with the road slips or skids, then the metal rim the tire is mounted on is affected the same way. The rim can't slip when the tire doesn't. Yet the rim slips while the tire doesn't is one "solution" to the paradox given on Wikipedia. It makes no sense except as a far-fetched metaphor. In other words, the rim "slips" but it doesn't really slip. The rim "skids" would be less far-fetched.

Aristotle's wheel paradox #1

Wikipedia. I judge the article as poorly written, especially when it says the paradox is about two wheels. A comment on the Talk page agrees. The quote from Mechanica, written more than 2,000 years ago, describes the paradox as about two circles. Two circles can depict one wheel, e.g. like on a car or truck, with the smaller circle depicting a metal rim. Or the two circles can depict one tire -- not mounted on a rim -- with the smaller circle depicting its smallest circumference, the bead or lip. They can depict a roll of tape.

If the rigidly coupled circles are rolled a full revolution, then all points on both circles have the same position relative to their common center at start and end. Every point's translation vector has the same direction -- parallel to the horizontal surface -- and length as the center's translation vector. Such length is 2*pi*R, where R is the radius of the larger circle. This necessary fact about translation elegantly solves the paradox. Every point on the smaller circle must move 2*pi*R. This shows that the smaller circle's circumference 2*pi*k*R, where k is its circumference divided by R, is irrelevant for one rotation and the given setup. How far the smaller circle moves horizontally is dictated by its center.