Strange Relationships of the Circle

In summary, the S/L and {[S/L]/R} ratios of a circle can vary as a function of the radius size due to the fact that S and L have different units and the ratios are dependent on the radius. There were also some errors in the provided tables, but these can be corrected using simple algebra or arithmetic.
  • #1
dom_quixote
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PHOTOGRAPHIC REDUCTION OR ENLARGEMENT
magno.JPG


The proportions of a circle never change. But...

radius.JPG


Question:
If a circle is always a circle, then how is it possible that the S/L and {[S/L]/R} ratios of a circle can vary as a function of the radius size?
 
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  • #2
Because they aren't unitless quantities. S has units of length squared and L has units of length, so when you double the length, they don't change the same way.
 
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  • #3
There are errors in your table for the perimeter. It should read 2/5π and 3/5π 2/3π for the first two. And S/L for R=1/2 is 1/4, not 1.
 
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  • #4
jack action said:
There are errors in your table for the perimeter. It should read 2/5π and 3/5π for the first two. And S/L for R=1/2 is 1/4, not 1.
##\frac 2 3 \pi## ?
 
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  • #5
dom_quixote said:
If a circle is always a circle, then how is it possible that the S/L and {[S/L]/R} ratios of a circle can vary as a function of the radius size?
Try it with a square if you have problems understanding it for a circle. It's more a geometry question rather than physics.

Related topic:
https://en.wikipedia.org/wiki/Square–cube_law
 
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  • #6
There are many basic arithmetic errors in your charts. The [S/L]/R column should always be 1/2, for instance.
 
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  • #7
In fact, I made a mistake :sorry:.
The corrected table is below:

rad_corr.JPG


Note in the TABLE I a singularity, when R=2/1:
S = L ?
Certainly not!
S expresses area;
L expresses length.

P.S.:
If there is another error in the table, I apologize for my numerical dyslexia :wink:!
 
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  • #8
How about applying some simple algebra before you make the tables?
Since ##S=\pi R^2## and ##L=2 \pi R## then ##\frac{S}{L} = \frac{\pi R^2}{2 \pi R} = \frac{R}{2}## and ##\frac{(\frac{S}{L})}{R} = \frac{1}{2}##.
That is all there is to this, now you can correct your tables.
 
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  • #9
dom_quixote said:
S = L ?
Certainly not!
S expresses area;
L expresses length.
They can have have the same numerical value (which is what your table shows), but different units (which your table doesn't show).

dom_quixote said:
If there is another error in the table, I apologize for my numerical dyslexia :wink:!
See post #6 and #8.
 
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  • #10
dom_quixote said:
Note in the TABLE I a singularity, when R=2/1:
S = L ?
Certainly not!
S expresses area;
L expresses length.
Why don't you use letters that more closely align with what they represent?
R is fine for radius, but why are you using S for area and L for length? Better would be A for area and P or C for either perimeter or circumference.
DaveE said:
How about applying some simple algebra before you make the tables?
Or even some simple arithmetic.
You have errors in the first two rows of table 1.
##2\pi \frac 1 5 \ne \frac{5\pi} 2##
##2\pi \frac 1 3 \ne \frac{3\pi} 2##
 
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  • #11
The initial questions have been asked and answered, so I'm closing this thread.
 
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Related to Strange Relationships of the Circle

1. What is the concept of "Strange Relationships of the Circle"?

"Strange Relationships of the Circle" refers to the unique and complex interactions between different elements of a circle, such as its radius, diameter, circumference, and area. These relationships are often counterintuitive and can be explored through mathematical equations and geometric principles.

2. How do these relationships differ from traditional geometric concepts?

Unlike traditional geometric concepts, which focus on straight lines and angles, "Strange Relationships of the Circle" examines the curved and circular nature of a circle. This allows for a deeper understanding of the interplay between different elements of a circle and how they affect one another.

3. What are some real-world applications of these relationships?

The relationships explored in "Strange Relationships of the Circle" have numerous practical applications, such as in engineering, architecture, and physics. For example, understanding the relationship between a circle's circumference and diameter is crucial in designing circular structures, such as bridges and tunnels.

4. Can these relationships be applied to other shapes besides circles?

While the focus of "Strange Relationships of the Circle" is on circles, many of the principles and equations can be applied to other curved shapes, such as ellipses and parabolas. These relationships can also be extended to three-dimensional shapes, such as spheres and cylinders.

5. How can studying "Strange Relationships of the Circle" benefit us?

Studying the unique and intricate relationships of the circle can improve our critical thinking skills and problem-solving abilities. It also allows us to better understand and appreciate the beauty and complexity of our world and the mathematical principles that govern it.

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