Stepping off a polygon

In a recent thread on another forum, someone wanted to make a 14 tooth ratchet and asked for methods for dividing a circle into 14 segments. Since he lacked a rotary table and dividing head, it was suggested that he use dividers to step off chords around the circle. After all, this was good enough for the ancients who made the Antikythera mechanism so it ought to work here.

It was suggested that he guess the initial divider setting and step off around the circle. If he didn't get 14 divisions, adjust the dividers and try again. Continue this iterative process until he obtains exactly 14 divisions.

Now, anyone who knows anything about trigonometry will realize that iteration is about the dumbest way to approach this problem. So how do we calculate the initial spacing of the dividers?

Refer to my canonical diagram of a circular arc below...

https://s3-us-west-1.amazonaws.com/f...e/111839-1.jpg

A regular (i.e., all sides equal length) polygon of N sides has been inscribed in a circle. The chord BD represents one of these sides. At the center of the circle it subtends an angle of 2x. We'll use the label T for 2x, i.e., T = 2x. Since all the sides are equal we know that T = 360/N.

We now have an isosceles triangle, ABD, where we know an angle, T, and the two equal sides, r, and wish to know the remaining side, BD. Fortunately, trigonometry provides us with a formula, called the "law of cosines", to compute this side. If 'c' is the unknown side opposite the angle 'theta' and 'a' and 'b' are the other two (not necessarily equal) sides, then the law of cosines says...

c^2 = a^2 + b^2 - 2 * a *b * cos(theta)

Applying this to our problem, we have...

(BD)^2 = r^2 + r^2 - 2 * r * r * cos(360/N) = 2 * r^2 * (1 - cos(360/N)

And, solving for BD, the required divider setting, yields...

BD = r * sqrt (2 * (1 - cos(360/N))

Now, as a check, every schoolboy knows that if you step off the circle with the radius used to draw the circle, you'll get a perfect regular hexagon. So, with N = 6 the formula should show us that BD = r. Substituting, we have...

BD = r * sqrt (2 * (1 - cos(360/6)) = r * sqrt (2 * (1 - cos(60)) = r * sqrt (2 * (1 - 0.5)) = r * sqrt (1) = r Q.E.D.

So, armed with this formula, you can compute the required divider setting for stepping off a regular polygon of any number of sides.


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A little math aside here. Note that if c is the hypotenuse of a right triangle opposite the right angle, theta = 90, the law of cosines reduces to the Pythagorean theorem, c^2 = a^2 + b^2

Not to be picky, but triangle ABD is an isosceles triangle, not an equilateral triangle, except in the one case where BD=r.

Stepping off would be faster than the time spent reading this alternative method, let alone trying to understand it or working it out. However, now knowing the math, it might be useful at another time.

I have never being very good at maths so you could try it my way, using CAD draw a circle of the required segments, radius and measure B and D. After that it comes down to how good you are at setting and using dividers.

There is nothing wrong with math, my trig started failing me years ago, always good to see the formulas posted. I am also guilty of over using cad to work out my problems instead of hunting down my scientific calculator.

Quote:

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Originally Posted by mattthie

Not to be picky, but triangle ABD is an isosceles triangle, not an equilateral triangle, except in the one case where BD=r.

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Thanks for pointing that out; I've corrected the original post.

Another way to dived a circle into the number of segments is to use a degree wheel or angle finder compass take the number 360° divide by the number desired note the degrees then set the compass to the desired angle on a center line extending from the center to any point on the circle draw lines from the center along the compass arm set the dividers points to the intersecting points then step off the points and your done. the only math required is simple bonehead division
Marv's way is a lot more eloquent if your not like me who's higher math skills have increasingly failed over the years from disuse

Quote:

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Originally Posted by Frank S

Another way to dived a circle into the number of segments is to use a degree wheel or angle finder compass take the number 360° divide by the number desired note the degrees then set the compass to the desired angle on a center line extending from the center to any point on the circle draw lines from the center along the compass arm set the dividers points to the intersecting points then step off the points and your done. the only math required is simple bonehead division
Marv's way is a lot more eloquent if your not like me who's higher math skills have increasingly failed over the years from disuse

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360/14 = 25.174... deg

A protractor capable of setting that angle accurately is going to cost a lot of money. It's a lot easier to set dividers accurately to a precise linear measurement.

I don't understand what the terror over a bit of math is all about. You simply plug numbers into one simple equation from my original post...

BD = r * sqrt (2 * (1 - cos(360/N))

and you have the required length for an N-sided polygon. Yes, a scientific calculator or equivalent is required but these are all over the web.

No argument there Marv, but I was 1 of those who still used my slide rule and Smoles to check my scientific calculator well into the 90s about 2000 I got my first computer in 2010 I suffered extreme oxygen deprivation which killed off much of the analytical portion of my brain, which has only recently been partially returning due largely to a lot of your postings

Quote:

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Originally Posted by Frank S

No argument there Marv, but I was 1 of those who still used my slide rule and Smoles to check my scientific calculator well into the 90s about 2000 I got my first computer in 2010 I suffered extreme oxygen deprivation which killed off much of the analytical portion of my brain, which has only recently been partially returning due largely to a lot of your postings

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Checking a scientific calculator with a slide rule? Good one, Frank, I'm still laughing.

Every shop should have a scientific calculator. I regard it as the second most important tool. Today, good ones, e.g. Casio, can be had for less than $20.

I have an HP35S in the shop and one at my desk. A bit more expensive but they're programmable and that's a big plus for me.

If you want to try one yourself, you can download an emulator here...

https://www.educalc.net/2336231.page

and the manual to go with it here...

https://www.educalc.net/2336445.page

Unusual for HP, the HP35S can be switched between RPN and algebraic mode, though why anyone would use the latter is beyond me. :-)

I am still using my old HP45 calculator that I purchased new in 1973 for USD395. (Wiki tells me that in 2017 terms that was the equivalent of USD2178).
It was the first scientific calculator that I had ever heard about, and I had visions of it changing the world.
I can read it in poor light, because unlike modern calculators it has a red LED display.
Apart from its speed advantage, I found that the RPN also had a distinct advantage in a work environment, because nobody ever asked to borrow it more than once.

Moby Duck,

I like your story about the HP-45. I remember it well. In 1973 I started graduate school for MS and PhD degrees in geophysics. The geophysics department owned an HP-35 and HP-45 calculators that we shared among 10 grad students. Other than using a Wang calculator (the one with the cool Nixie tubes) in an undergrad statistics class, I never used an electronic calculator before 1973 and only used a slide rule starting in 1962 (and still know how to use mine). Wow, what a world of difference these wonderful HP calculators made for our work. By 1975 the cost of simplest electronic calculators had become affordable and I bought one at the local Kmart store. Years later I bought an HP-65 programmable calculator with storage for 100 keystroke instructions (in its time this was a big deal) and still have the calculator as a museum piece. Almost all of my grad work was preformed with computers using FORTRAN code for research but the HP calculators were great for double checking our work.

Regards,
Paul

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"Stepping off would be faster..." Any student with grade 11 math should have no trouble applying the cosine law to a simple diagram - which anyone should have no trouble sketching...not much understanding required.

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Originally Posted by volodar

"Stepping off would be faster..." Any student with grade 11 math should have no trouble applying the cosine law to a simple diagram - which anyone should have no trouble sketching...not much understanding required.

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I'm inclined to agree with you.

Nevertheless, most posts involving anything remotely mathematical beyond the four functions will have posters with little on-topic to offer advertising, as if proud of it, their ignorance of mathematics.

How is it that the educational system(s) have failed so miserably to instill an understanding of elementary math below the calculus level?

Everyone will have his own answer to that question so I'll offer my own thoughts on the subject to get the ball rolling.

From my own experience, teachers fail to answer what the first slide in any presentation should be: Why do you want to learn this? Admittedly, in math it's difficult to answer if the student doesn't yet know how to use what's being introduced but, by re-answering the question as each bit of new knowledge is presented, some degree of communication can be effected.

Still, the student has to apply himself and learn. I think a large part of that problem stems from the fact that math is the first time he is faced with a subject that is absolute, offering no opportunity for interpretation. You can argue endlessly about what the author of Catcher in the Rye meant or what the impact of the North's victory at Vicksburg was on the Civil War but the solution to a polynomial equation provides no opportunity for self-expression. Students denied the opportunity to manipulate the subject react by mentally retreating from it. (OTH, they could just be typically lazy kids who don't want to learn anything. :-)

Finally, a society that denigrates intellectual achievement makes it difficult, bordering on anti-social, to demonstrate learning. Sports achievement outweighs any learning that might be useful in the world most students will enter.

Quote:

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Originally Posted by mklotz

From my own experience, teachers fail to answer what the first slide in any presentation should be: Why do you want to learn this? Admittedly, in math it's difficult to answer if the student doesn't yet know how to use what's being introduced but, by re-answering the question as each bit of new knowledge is presented, some degree of communication can be effected.

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Agreed. They have also failed to address the reasonable question posed by non-technical students: "When will I ever use this in my life?"

When I hear kids asking this, I say: "If you're asking that question, the answer is that you will probably never, ever need to use this specific bit of mathematics in your entire life." This backs them into saying: "Then why am I wasting my time learning this?!"

Then I explain that, for non-technical people, the benefit of math is not necessarily the math itself, but the problem-solving techniques:

-Acknowledging the existence of a provable, immutable, and universally-accepted answer, devoid of nuance and opinion.
-Using knowledge as stepping stones, where each level builds upon the previous one.
-Understanding that some components of life are mechanistic and deterministic, and you'll never change that.

And the most important:

-Doing something difficult, hating every second of it, and completing it anyway.