AP Wheel Geometry

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John Collins
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re: AP Wheel Geometry

Post by John Collins »

I’ve only watched the video once Silvertiger but I congratulate you, and I must watch it a few more times to take all in. Amazing job!

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re: AP Wheel Geometry

Post by raj »

Mountain versus Mole-hill???

Septagram = mirage ???

360 / 7 = 51.42857142857143 degrees ( 7 equal angles), which would not fit EXACTLY in a circle.
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re: AP Wheel Geometry

Post by daxwc »

Great work Silvertiger. Boy drafting and engineering have changed since I was taught them.
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re: AP Wheel Geometry

Post by ME »

Silver... impressive demonstration.

But it looks to me your are checking your own hypothesis against your own geometry....
and then of course things will fit perfectly!

That angle, for instance, can't be 25.7143° from measurement.
It's more likely that value is the geometrical exact fraction: 25 ⁵⁄₇.

You simply can't reach such accuracy from measurement. You literally (mathematically) need an image of a million pixels wide (which I seriously doubt you have), and then still hope to find the best fit for such best possible accuracy. Even when you average out the print-press, woodcut errors and photographed page-curl-deformation mistakes.

When you fit a 7-star (radius=pi/7) on a regular-15-polygon (radius=2·pi/15) as you did, the radius ob which the inner 7-star-point rests should be exactly 1/15th of the outer radius of the polygon.

According to the AP-wheel that 7-point should fall inside the inner black circle.
Curiously it's radius is optically less by about the same amount the next concentric circle is larger. So there may be still hope, but there is no perfect fit as shown/intended.
When stretching things to make it fit again then you get your demonstrated 55 inch, but it's a conclusion from your own AP-inspired geometry; but not implied from the AP-wheel.

I measure the outer circle (where the sector-base meets) as 9.86(±0.06)* times the radius of the inner circle (where the sector point meets)
With this the star should have 6.74 points.
Or, the 'real' angle of the AP-sector should be around 26.68°(±0.02)
* nb. This according to a current calculated best-fit, and a recentered center (the AP-wheel is not symmetric)... and show pixel-fit accuracies, but things are actually worse on the AP-wheel drawing.


TL;DR;
I'm just saying here that the "fit" is not as precise as presented, but 7 is still the amount closest to 6.74.
With all this these demonstrated septagrams may still be as intended by Bessler.
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Post by Silvertiger »

ME, go to the very first post of this thread and look at how I measured it, took the averages, how they add perfectly to 120, and how I got my percent error at less than one percent.
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Re: AP Wheel Geometry

Post by ME »

Are your really serious ??
Silvertiger wrote:1. The small angle [A]: (25.0553 + 25.9897 + 26.8006) = 25.9485
2. The large angle : (92.9559 + 93.8318 + 95.3667)/3 = 94.0515
3. Together: 25.9485 + 94.0515 = 120 degrees exactly

Sum of angles in a circle = 360 degrees (per definition) in three parts is 120 degrees exactly.
With 1 percent accuracy your sum of angles is (360±3.6)/3 = 120±1.2

Also for a precision of 0.0001 degrees:
ME wrote:You literally (mathematically) need an image of a million pixels wide (which I seriously doubt you have)
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Post by Silvertiger »

Well...I'm definitely not green when it comes to reverse engineering, as I have been doing this since '97. Those are averages from the measurements taken. (I did accidentally leave off writing in "/3" in the first equation for angle A.) This accounts for aberrations in printing/cutting, akin to reverse engineering, say, a wrecked car for example, but far simpler. AutoCAD does a wonderful job of scaling up and preserving raster images, and all you have to do is point and click and drag a line until it sits in parallel center of a line with a large lineweight - you pick a start point and end point. Pretty simple, yet very accurate. It's not nearly as hard as you seem to think. For example, a line drawn with a thick charcoal pencil will have abnormalities, and even more if copied and printed, etc. But just imagine that a thick line is actually a long rectangle. Well, then, all you have to do at that point is draw a line from the midpoint of one end segment to the the midpoint of the segment at the opposite end...and that's it. And the fact that I have three chances to measure one angle gives me the accuracy I need.

Those are averages taken from the interior angles, and they are accurate. 25.9485 degrees is the average for the white area in the AP wheel; 94.0515 degrees is the average for the black. Both averages added together yield a sum of exactly 120 degrees. On the first angle, A, in which the measured average is 25.9485 degrees, I compare it to what is expected for a heptagram fit of 25.7143 degrees. When calculating percent error, I get 0.91%, which comes to a variance of only 0.2361 degrees. On the second angle, B, in which the measured average is 94.0515 degrees, the expected heptagram fit angle is 94.2857 degrees, yielding a percent error of 0.25%, which comes to a variance of only 0.2351 degrees. How do these small variances translate to ±1.2? Please think critically for just a smidge longer before responding this time lol. :)

Edit: Addendum - the image I used is very detailed. I got it from Oystein's website. I've attached it below, and you can see that it is detailed enough that it can be scaled up quite nicely without having to convert it to a vector image. I have also included an image for the tracing method described above in sketching a rectangle to match a lineweight and then drawing a line down its center from midpoint to midpoint. However, I used a slightly faster method in which I drew one end segment and then traced down the line using point projection to a point significantly far past the opposite end, positioning the point until center alignment was achieved. This is highly accurate as well, and is akin to surveying property lines using a point anchor and a length of string. In other examples, the same accurate method is used in CAD when surveying properties or when measuring square footage of roofs (given the pitch) from "Eagle Eye" satellite images.
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AP Wheel.jpg
Lineweight Rectangle.jpg
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re: AP Wheel Geometry

Post by ME »

My post was actually meant as a small side-note, and it's not about a typo.

360° simply makes a full circle.
Take any circle, divide it in as many irregular sectors (through the center) as you want. Then measure the angles. Add them all up. You'll find 360 degrees.
Average by three: 120.

For that AP-wheel I measure a diameter of 392 pixels.
That makes a circumference of 1231.5 pixels, so with a precision of 1 pixel (half pixel to whichever side) your accuracy of a single line will be around 0.29 degrees when measure from the center. It's 0.58 degrees when both the start- and end-point wanders.
A sector is between two different lines: 0.58 degrees
When measure between two wandering lines: 1.17 degrees

Example:
Let's measure the top-right sector.
You cornered it between two lines

The upper with (dx;dy)=(228;-212)
The lower with (dx;dy)=(331;-96)
Let's say the chosen lines are as accurate as possible, there's still a positioning error of one half pixel for choosing both the starting point as the ending point of that line: the precision.
The angles become:
Upper: 42.9174°(±0.260)
Lower: 16.1738°(±0.206)
So the sector width is 26.74°(±0.47)
It fits a star-figure with the amount of points between: 6.62 ... 6.85 (still roundable to 7)

...nitpicking perhaps. But it seems important. Especially with your motivation:
silvertiger wrote: The septagram is not just a best fit...it is nearly perfect, and that is accounting for percent error as well. Precise numbers are derived from averages every day. It gives us predictable models in statistics and allows precision and accuracy to be met in experiments. We observe, and we test. We record, and then we model the results. I like geometry. That's why I did this.
Here's my measurement attempt (I thought it would fit better there): http://www.besslerwheel.com/forum/download.php?id=18492
Many, but not all, of our lines are almost on top of each other. But your reported angles are (arbitrarily?) between the angles of the rim-division and the angle of the triangle/sector.

I actually wanted to go to this point:
Silvertiger wrote:If Bessler wanted the corners to meet, he would have adjusted the angles...but he didn't.
The same for the radius where the septagrams could meet, when the corner match. He could have drawn a smaller inner circle... but he didn't.
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re: AP Wheel Geometry

Post by rlortie »

Septagram:
One of the oldest recorded meanings given to this star may be found within Kabbalistic tradition, where it represents the sphere of Venus and the power of love.
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Post by Silvertiger »

ME, look on the bright side...we both measured it using different methods. I do believe that yours involved much more work than my surveyor's method. That is excellent work. Without having CAD software, very impressive. However, I fear having CAD software and pertinent training has given me a slight unfair advantage...and I do mean slight, not much at all. :D

Your own measurements of the black area angle yield an average of 93.9133 degrees, a difference of only 0.1382 degrees from mine. For the white you measured an average of 26.09, a difference of only 0.1415 from mine. These numbers are GREAT! Your percent error is slightly higher than mine, but the numbers are so small that yours is also an accurate trace. Your percent error for the black area angle is only 0.4%, and 1.44% for the white. Yeah, it's just under double what mine is, but just think, you achieved this level of accuracy counting pixels! The sum of your averages is 120.0033, which is awesome. If anything, this serves as a confirmation that the white area interior angles are indeed the interior angles of a heptagram.

As far as the circumference of the axle, there is a "play" of one inch due to the thickness of the line, aka the "lineweight". I averaged it out to be a four inch radius. So I was not surprised that it was within the range of play when I inscribed the dodecagram to get the axle radius of 3.763 inches. I just tried using some logic for that one: since the dodecagram is ever present, how could anyone who traces it out NOT see the implied axle? Although it is a guess, it is an educated one based on two data sets: one of measurement and one of implicit geometry.
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Re: re: AP Wheel Geometry

Post by ME »

Oh thanx! I was not competing, I think.
These are my values with more decimals: [26.38719872; 26.53920356; 25.34412344; 94.94795137; 93.42529610; 93.35622681]
rlortie wrote:Septagram:
One of the oldest recorded meanings given to this star may be found within Kabbalistic tradition, where it represents the sphere of Venus and the power of love.
Well, in that case:

Free mason --> Vesica Pisces (and 1,2,3,4,5) --> The Tree of Life -->A heptagram --> Alchemy --> wiki: Vitriol --> Magic squares & Planets

video: Heptagram construction --> wiki: Albrecht Dürer --> woodcut prints, Nuremberg -->Item 5: Nuremberg globes...?
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re: AP Wheel Geometry

Post by daxwc »

Albrecht Dürer is always an interesting subject to study.
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re: AP Wheel Geometry

Post by Dave Roberts »

The discussion of the angles is very interesting but there may be other factors not considered. I may be called out on this for referencing Alden Park and his "Interpretation of Bessler's Little Book Parable" but will the following excerpt apply:

...Near full strength power production for the working pendulum, the major shadow boxing begins when the center of the small wheel's angular amplitude is at a lower shaded edge of one of the upper light sectors of that circle that I drew for you at the end of this Apologia. Please note that after the shadow boxing strike which stops the acrobatic turns of the torque governor wheel, the always turning little wheel's center angular amplitude curiously increases from its stopped position (in apparent defiance of gravity) to near the top of the white sector of my figure before stopping, returning, and going down through the shadowed region in my figure. When initially not yet at full strength of power production, the two upper unshaded sectors should be drawn lower (though not at the same height) so that their lower portions coincide with the main pendulum line angles of expected-maximum main-pendulum-line angular-amplitude for the center of the small wheel....

This appears written by Bessler. Comments?
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re: AP Wheel Geometry

Post by daxwc »

Silvertiger could I please bother you for the original circumference in your video 8)
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Post by Silvertiger »

Which circle?
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