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December 21st, 2014, 01:53 AM   #1
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Math Focus: Non-soluable tasks
Regular polygon

Dear cleverminds,
May I use mechanical proof shown below for newly invented POLYCOMPASS
(mathhelpplanet.com/viewtopic.php?f=548&t=36796), which is most simple way to build any POLYGON ?
========
Let's have a look into simple similar construction - 2N hinge-crossing sticks REMOTE ARM, which looks like XXXXXX . When system is open it can move each middle hinge at same time in sequence for distances H/2 , 3H/2 , 5H/2 ….. ((2N-1)H)/2 appropriately (H is stick height). Being bended into the ring , it changes the movement nature . No linear steps are possible since the ring is locked, simultaneous radial expansion only. There are two ending positions with two values external circumference radius R: sticks are vertical ( R→0) and horizontal (Rmax). As N→∞ , circumference length varies between 0 and NH, so Rmax ≈ NH/2π.


HOW TO BUILD REGULAR POLYGON WITH COMPASS AND RULER

How to cut out the plate in order to obtain regular polygon with desired number of edges?..
The problem can be described another way: how to divide a circle for N equal parts, where N is an integer. First of all, the question arise: for how many equal parts theoretically the circle can be devided with a compass and a ruler? This question has been answered ultimately by K.F. Gauss : not any number.
You can : 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 17,..., 257, ... more parts.
You can’t: 7, 9, 11, 13, 14, ... parts.
Besides, there is no common way of drawing . Section for 15 parts, for example, is not similar as for 12 parts, etc. It is very difficult to keep all ways in mind.
Division of a circle for N equal parts, one of the oldest problems in mathematics; is to produce d. c. using only a compass and a ruler. Ancient Greek mathematicians were able to devide the circumference by 3, 5, 15 pieces, as well as the doubling of the number of parties unlimited of polygons. In 1789 K.F.Gauss has proved that a circle can be devided with a compass and a ruler on 17 parts and in general on the number of parts (N), which can be represented as N= 2^(2 k + 1) and is a simple or is the product of these numbers and any powers of 2 (when k = 0, 1, 2, 3, 4 are prime numbers N = 3, 5, 17, 257, 65537. If k = 5, 6, 7 the relevant number is not prime). You can’t do it for any number of equal parts with simple compass and ruler. D.c. task is equivalent to solving hyperbolic equations x^(N-1) = 0. D.c. with a compass and a ruler is only possible when all the square and linear equations roots are available.

As Gauss persuaded THERE IS NO WAY, so no one else tried… . But it is possible if to devide two circles at same time by modified compass, which consist of more than two sticks and more than one hinges.
May we do that ? Why not ?

So we can create more and more complicated ones.
The only thing we have to legalize is CROSS HINGE with no any compromises against Euclidean geometry postulates. It is self-explainable thing that all hinges are ball-type.
Hyperboloid is well known unique body built of straight sticks.Pic1.


No matter how to turn upper circle, clockwise or anticlockwise. Turns with equal angles built same bodies as well. Let’s unite two in one.Pic2.


Very easy job.
2 N equal sticks are taken. We can find the middle of each one by means of “normal” compass and drill their ends for the hinge holes.

Let’s place N sticks like these.Pic3.

Let’s place another N sticks across as it’s shown below .Pic4.

All O-marked dots are to be hinge connections.
Whole system has to be twisted into the “whirl” as it is shown by blue arrows and its connection completed .
System turn angle and stick length must provide an odd number of crosses (3 in this case) to obtain the MIDDLE LINE. That line is forming ABSOLUTELY SAME diamonds . Such condition is the reason of polygons regularity as desired result. Moreover, the whole system may vary its diameter keeping regularity itself!
This is close to 0-diameter pentagonal formation. Pic5.

The same regular pentagon after “pulling out”.Pic6.

No doubts, all circumference section N marks are pointed by hinge axis ends .
By means of such POLYCOMPASS we can draw any regular POLYGON.

Valeriy Bryukhovets
Master Mariner
Author of “COMPUTERIZED ASTRONAVIGATION” (2011-Reebest-Sevastopol-Ukraine).

Last edited by Val Bryux; December 21st, 2014 at 02:30 AM. Reason: Article English version
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December 21st, 2014, 11:47 AM   #2
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Do you have a question?

And why your link?
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December 23rd, 2014, 06:21 AM   #3
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Math Focus: Non-soluable tasks
The question is above the article....
=MAY I USE... ?=
I'd like to use another construction (simple analogy) to prove.
Read thoroughly, please. Link (russian version) is given to see the pictures.
All pictures are numbered in order
of appearance in original article.
It seems, there's nothing difficult for specialist....
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December 23rd, 2014, 09:05 AM   #4
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Well, I'm leaving it for Tahir...he's an expert with such clear problems...
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