personally i think all angles can be constructed using ruler and compass only. if not then i think i have discovered a method and if any one in capacity to help me publish it can just contact me.
2007-01-31 20:25:31 · 7 answers · asked by mustang nigel 1 in Science & Mathematics ➔ Mathematics
Even if it's been discovered, and you just have a different way of showing how it can be done in a finite amount of steps, I'd publish it anyway. Try writing it up and submitting it to a few different math journals to see who accepts it.
2007-01-31 20:51:57 · answer #1 · answered by Anonymous · 0⤊ 1⤋
You said ruler and compass. That is different than straight edge and collapsable compass which is what the ancient Greeks used in their construction techniques.
The collapsable compass could not be used to measure a length and then be picked up to recreate the same length elsewhere. As soon as it is picked up it loses its measure. Euclid probably did that to create a minimum of pre-existing conditions. Also so that the entire construction would be shown on the page. If you measure a length and then duplicate it elsewhere on the page then it would not be evident to future observers of the page that two lengths are necessarily equal. The evidence was only in the compass, not on the page.
As a practical matter, a length can be transferred from one portion of the page to another thru a certain technique anyway, even with a collapsable compass. So that is not a real concern.
But the use of a ruler, or making a mark on the straight edge with the compass is forbidden in classical constructions. Obviously the use of a ruler or marking a straight edge would allow some additional constructions, but they would be made using different rules.
Under the classical conditions it has long been proven that certain things cannot be done. Most notably
1) You cannot trisect an arbitrary angle.
2) You cannot double a cube. Meaning that if you have a line of length one, you cannot create another one of length 2^(1/3).
3) You cannot square a circle. You cannot create a circle and a square of equal area.
It has also been proven that only certain regular polygons can be constructed. Carl Gauss showed that regular polygons can be constructed that have the number of sides
2^(2^n) + 1
if that number is prime.
As far as anyone knows, only the first five numbers of this sequence are prime. But that has not been proven. Starting with n = 0 this yields polygons with sides numbering
3, 5, 17, 257, and 65537
Obviously also, regular polygons with certain multiples of these, but not all multiples, can also be constructed.
2007-01-31 21:23:47 · answer #2 · answered by Northstar 7 · 0⤊ 0⤋
I recommend you ask a professional mathematician to review it.
However, it has been shown that you can not construct a 20 degree angle. This has been proven, so I am very skeptical.
You can approximate any angle you want to any degree of precision, though. You can bisect an angle, so describe the angle in binary as a binary "decimal" times pi. Take a straight angle, and bisect it.
Then, if your binary number is >1/2, bisect the angle between pi/2 and pi radians. If not, bisect the one between 0 and pi/2 radians. You continue in this fashion, bisecting the angle on the clockwise side of the bisector you last created if the digit is one, and the other angle otherwise, and move right one digit. If you do this infinitely many times, you have any real angle. The sequence of lines converges to the angle. You will never get there, though, unless it is divisible by two.
You can construct any angle A iff pi/A is a rational number, and the numerator of the fraction, in the most simplified form, is the product of unique primes of the form 2^n + 1 (no two of these factors can be the same) and an arbitrary number of twos. This is because the Mersenne Prime-gons are all constructible on an arbitrary circle with a predefined point.
2007-01-31 21:17:54 · answer #3 · answered by a r 3 · 0⤊ 0⤋
All multiples of 3° can be constructed by ruler and compass, but it is not possible to construct an angle having any other integral number of degrees. For example, it is not possible to construct a 40° angle using ruler and compass, because 40 is not a multiple of 3. There is no simple way to classify all possible angles that can be constructed with ruler and compass.
2016-03-28 23:26:09 · answer #4 · answered by Shennen 4 · 0⤊ 0⤋
Mathematically, a straightedge can only find linear points, and a compass quadratic points (from the equation of the circle). While it is not a simple proof, it has been shown that the only angles of finite order that may be constructed starting with two points are those whose order is a product of a power of two and a set of distinct Fermat primes.
This is explained in detail at http://en.wikipedia.org/wiki/Compass_and_straightedge .
It is quite possible that you have some construction using other properties than those allowed in the traditional problem. Many problems that are impossible, like trisecting an angle, can be done with only slight extensions.
2007-01-31 21:20:04 · answer #5 · answered by sofarsogood 5 · 0⤊ 0⤋
No, it is easy to see that you can construct onlu those angles whose sin,and tan are algebraic numbers. Try you rmethod with
sin x =1/pi.
2007-01-31 20:39:58 · answer #6 · answered by gianlino 7 · 0⤊ 1⤋
you can construct 60 its multiples and sub multiples like30,15,7.5 ....or 120 you can add any of them to other to get many angles
2007-01-31 20:35:58 · answer #7 · answered by Anonymous · 0⤊ 0⤋