AP Wheel Geometry

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Silvertiger
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AP Wheel Geometry

Post by Silvertiger »

I made this thread to get it out of the way of Oystein and Stewart's discussion. This thread is for John Collins, and anyone else who is interested in these findings.

Ok, so Oystein has a pretty detailed scan of the AP wheel on his website. I dropped it into AutoCAD and traced it by hand after scaling the outer rim to a radius of 36 inches. I traced every single line straight down their respective centers, as they have a certain lineweight or thickness to them. Then I dimensioned every single angle, as the drawing has a certain level of imprecision, but yet very precise for its day. Now let's take the average of all angular dimensions:


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


Image






4. Adjusting to 26 and 94 degrees...the pentagon does NOT fit, as the corners do not meet.
If Bessler wanted the corners to meet, he would have adjusted the angles...but he didn't.




Image
Image







5. The closest polygonal fit is a septagon, where A = 25.7143,
and B = 94.2857 degrees. Sorry guys, this is where the math takes it
.



Image
Image







Here is the outcome:



Image





John Collins wrote:@Silvertiger, I credit you with the accurate assessment of the measurement of the Apologia wheel, but I feel that you may have taken the drawing measurements too literally.

It has always seemed apparent to me that the printing technology was not capable of the kind of accuracy you are depending on. As I explained on my web site, I measured the angles at the inner end of the white segments and discovered, as others have found, that the angles are ambiguous – a bit too vague to measure accurately. I noticed that the angles in the white segment formed a point outside the inner circles and that the black segments did not in fact form any measurable angle unless you extended them to a point which came somewhere beyond the centre of the wheel.

I measured the white angles again and established that they were variously somewhere between 23 degrees and 27 degrees. I added together each set of the same three numbers forming each of the three angles to see if the sum of the three numbers had any meaning. Using the angles as measured between 23 and 27 degrees, I ended up with several possible totals between 69 and 81.

Only one total was significant and that was 72 (3 x 24) which is we know a significant indicator of a pentagram. We also know that a pentagram is easily detected in several of Bessler’s published drawings. I found other supporting evidence that the Apologia wheel contains a pentagram, but that is all, in my opinion.

Adjusting your degree assessment would give 3 x 24, + 3 x 96 degrees.

JC

I didn't depend on accuracy John. That's why I measured every single angle and took the average, which is closest to the angles of a septagram. It's that simple. The confirmation of these numbers and their fit to the septagram is the fact that the averages themselves, without rounding, yield an exact sum of 120 degrees. Think about what that means in and of itself. Moreover, whether you round those averages to the nearest whole numbers of 26 and 94 degrees, or round them to the nearest polygonal fit, the pentagon still doesn't fit...either one. Therefore, I pretty much have to hold to the contention that the pentagon was not a shape that was implicit in the AP wheel whatsoever. The averages of these angles yield a very precise measurement of the interior angles of the septagram, a more accurate fit than the pentagon and 72. It is no different than applying an equation for best fit in Excel to points on a graph from a data column. 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. I really don't care what anyone thinks is represented or coded within the AP wheel to be honest...I just wanted to model the geometry and see where it led in a way that provides facts from a testable, scientific and mathematical approach as a viable alternative to conjecture. One can take it or leave it. Just try (and this applies to anyone who acknowledges that vision is usually colored by personal bias) not to get caught up with playing favorites. If I am picking something simply because I like it better than something else I see, I am probably shopping for a new truck or home furnishings. Other than that, I stick to what can be tested and verified.
John Collins wrote:I think that Bessler designed the drawing to hint at the pentagram. The white angles come to a precise point whereas the black ones don't. Therefore I think there is a temptation is to measure the whites first.

There are three of each so a measurement of each angle gives an approximate figure, but instead of averaging them as you have done, which might not provide the exact number Bessler intended, I sought a total whole number composed of three of each of the numbers between and including 24 to 27. The totals were 72,75, 78 and 81. I divided 360 by each total and there was only the one, 24 degrees, which divided into 360 exactly...5 x 3 times. I admit I assumed that it referred to the pentagram and subsequently discovered the intention behind the seemingly unnecessarily complex axle.

JC

John, it doesn't matter. The angles themselves don't match. 24 degrees to 26 degrees is too large of an error to ignore. I can draft free-hand to better accuracy than that. The 72 degree angle does not apply because the line drafted by its third of 24 degrees is way off as well. Whether you trace the white lines or the black lines, the angles still average out to be exactly the same. A missing corner makes no difference to an angle. Consider the percent error based on accuracy and precision from a best-fit model. In this regard the pentagon is the least likely candidate for best fit. The septagram, on the other hand fits the data perfectly, from which the angular deviation from the expected geometrical outcome comes to only 0.2342 degrees. This deviation is EXACTLY 0.2342 degrees for both angles. In science, that is extremely accurate. Here is the data:

For angle A: |25.9485[measured] - 25.7143[expected]| = 0.2342

For angle B: |94.0515[measured] - 94.2857[expected]| = 0.2342

%Err for angle A: 0.91%

%Err for angle B: 0.25%

Also, I thought more on what you said John, and I decided to work backwards, beginning with a pentagon to try to force the corners to meet. Turns out that the only way it works out is by using the angles of 27 and 93 degrees. These are the ONLY angles that can be drawn from center, having the center corners stretched out to any axle radius you want to pick (I measured 4 inches so I went with that), and arrayed about the center by separations of 120 degrees. So...it still doesn't work, and is by far the worst model fit having the largest percent error of all three models:
Image


According to the above data, the septagram's interior angles are the best and closest fit. Once again, the percent error for this fit is less than one percent.
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re: AP Wheel Geometry

Post by Silvertiger »

I just attempted a forced design whereby a pentagon would fit with inclusion of the 24 degree angles. However, the results are a disaster, as forcing the fit inevitably changes the larger angles; thus the model fails. The only way I can think of off the top of my head to keep all angles intact according to JC's vision of it is to actually move the axle of the wheel below center, which would still make it a forced fit and not a natural fit.

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re: AP Wheel Geometry

Post by raj »

You may be well right with your measurement.

I have only to take into account that Bessler's AP drawing is from hand-drawn woodcut.

As such, it may be lacking in clockwork precision.

I am inclined to go for (small angles 25 degrees X 3 = 75 degrees) + (large angles 95 degrees X 3= 285 degrees), to give an exact six-pointed star.

By the way, my own Auto Wheel design is based on two combined six-pointed stars, into a 12-pointed star.

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re: AP Wheel Geometry

Post by John Collins »

I get your point Silvertiger, but for me there are two problems with your diagnosis. I can’t see why Bessler would introduce a septagram, of course it maybe there is good reason so far unknown, but he doesn’t allude to one AFAIK - and secondly the pentagram is ubiquitous throughout his works, along with numbers 5 and 55.

I think that my diagnosis is more likely because accurate measurement of the angles was not necessary neither do I accept that he intended an average of the numbers involved when a simple calculation would provide the pentagram.

But I accept that you make a good point.

JC
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re: AP Wheel Geometry

Post by ovaron »

Good points, silvertiger. Without bias you measure 24 and 96 degrees. Also, you can not say on the one hand that Bessler worked very accurately and made no mistakes, but on the other hand then ignore 2 degrees and say that at that time you could not work more precise. If one assumes that Bessler has given a hint in his drawings, then the 7 is conspicuous. The 7 can be read from his portrait (hand position, right 2 fingers outstretched, left all 5), 7 buttons (3 on the right sleeve, 4 in the middle).
In "Gründlicher Bericht", the division of one foot has 7 divisions instead of 6. That can not possibly be a mistake.
And now in the APO wheel (excellent discovery, silvertiger)
Add to that the testimony "about 8 weights".
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Post by ovaron »

sorry meant 26 and 94 degrees :-))
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re: AP Wheel Geometry

Post by Oystein »

Seems you are may be mistaken about this?

You seem to be measuring the gap itself, not the area of 360 degrees needed to attach the dots at rim, is that right? Are you able to fit four of the white into a black?
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Re: re: AP Wheel Geometry

Post by Silvertiger »

John Collins wrote:I get your point Silvertiger, but for me there are two problems with your diagnosis. I can’t see why Bessler would introduce a septagram, of course it maybe there is good reason so far unknown, but he doesn’t allude to one AFAIK - and secondly the pentagram is ubiquitous throughout his works, along with numbers 5 and 55.

I think that my diagnosis is more likely because accurate measurement of the angles was not necessary neither do I accept that he intended an average of the numbers involved when a simple calculation would provide the pentagram.

But I accept that you make a good point.

JC
Just because you can't see it doesn't make it go away. People usually see what they want to see. You yourself have proclaimed your own measurements of the interior angles and used them as empirical data to support the pentagon fit. If you're throwing my measurements, out, then you must also throw yours out, otherwise it is simply straight up personal bias. I simply did a more thorough and accurate measurement using the law of averages to account for print aberrations. The numbers don't lie. I verified those angles as they miraculously add up to 120, even to four decimal places. The pentagon fit, once again, requires interior angles of 27 and 93 degrees; not 24 and 96. That is a HUGE gap. Have you seen Bessler's portrait. Did you see his drafting tools. He has more hand draft tools than I do, and I went to school for it. He knew how to draft an accurate design. Three degrees is a major flaw to consider. I had less than one percent error with the septagon fit. You can't just ignore it because you don't happen to like it. I don't know why he chose a septagon, nor do I care. I'm just showing what is empirically in front of us.
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Re: re: AP Wheel Geometry

Post by Silvertiger »

Oystein wrote:Seems you are may be mistaken about this?

You seem to be measuring the gap itself, not the area of 360 degrees needed to attach the dots at rim, is that right? Are you able to fit four of the white into a black?
Yeah, you're talking about the line segments. I traced the entire drawing and measured everything. Just go back to the top and look. The results are accurate, and the arc lengths are not 1/15th.
Last edited by Silvertiger on Fri Nov 03, 2017 12:56 pm, edited 2 times in total.
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re: AP Wheel Geometry

Post by raj »

Instead of my previous post, I assume that the small angle as 24 degrees and larger angle 96 degrees, then a pentagon fits in neatly.

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Post by Silvertiger »

Raj, no, it doesn't fit. Read the first post before posting please.
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re: AP Wheel Geometry

Post by Oystein »

What is the angle?
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re: AP Wheel Geometry

Post by raj »

Both my posts above, are assumptions, not facts.

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re: AP Wheel Geometry

Post by Silvertiger »

Here is your drawing the way you want it Oystein. I drew it based on 15 segmented divisions. Like I said, 27 degrees is what is required to make the pentagon fit. The drawing itself was measured at 25.9485 degrees, only 0.2342 degrees deviation from the interior angles of a septagram and thus at less than one percent error. Compare that to the whopping TWO degree error that exists with the pentagon fit.



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re: AP Wheel Geometry

Post by Oystein »

No, I asked the angle, when the opening is measured from the center of the main circle! Still talking of two different things..
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