/sci/ - Science & Math

>>6510254
that link is cool, it deserves a bump

bumping for cool

>>6510254
Well, it is cool.

Sacred Geometry Taught in a Donald Duck Cartoon:
www.youtube.com/watch?v=kVTPwPh7ioU

Why is this allowed?

>>6510254
thanks anon.

>>6511355
they are extensions of lines already created

>>6510254
my retarded attempt at a pentagon..

i just dont know what went wrong . jpg

>>6511939
ohh wow i finally got the pentagon.. 16 moves.. piece of shit i think it is bugged.. how the fuck am i going to do one less step this is already past the limit of my mental ability. I will be pissed if it is some obvious shortcut I wasted like an hour on this

>>6512195

>>6512217
sideways trick to use the extra line.. damn... damn thats slick rick

>>6510254
okay but how do you do the three circle pack in the origine circle ?

Hmm. Extra credit.

>>6511318
That has nothing to do with sacred geometry (some religious shit). It's Donald in Mathemagic Land or some shit.

This is a pretty awesome website, OP. I've actually invested some time learning to apply constructionist methods, this will be fun to do.

>>6511939
Pentagon is actually pretty hard. You may want to read up on some of the constructions for the golden ratio (in this context often referred to as the "mean and extreme ratio").

>>6515176
You can pretty trivially take your favorite N-gon and make a 2N-gon.

I should stop.

>>6515203
In case you're not already aware, there's a famous theorem proven by gauss that characterizes exactly which n-gons are constructable (it's not possible to construct just any n-gon, like a heptagon).

>>6515218
I was just looking that up. Shit's bananas.

>>6515220
Yea, it's kind of a shame that so many students come out of an undergrad degree in math thinking constructionist geometry is just a small sidenote in Algebra.

>>6515203
Well, shoot, it was starting to look like an N-gon where N was divisible by 3 would take 5N/3 moves to complete (3 took 5, 6 took 10, 12 took 20), but I was able to get a 48-gon with 79. Hum.

>>6515228

wat makes this constructionist?

I've done the circle pack 4 a bunch of times in different ways and it won't accept it at all. Any ideas?

>>6515249
You only have certain lengths and angles and you assert their existence by constructing them from the unit length. There are many lengths and angles that don't exist in this type of geometry (cube roots, for example) because they cannot be constructed.

http://en.wikipedia.org/wiki/Compass_and_straightedge

Addition, subtraction, multiplication, and division of unit lengths is possible as well as a bunch of field extensions using geometric techniques. Angles are a bit more complex but there are also many things that are not possible, like trisecting an angle.

On a historical note, there were all sorts of ideologies mixed up with geometry and mathematics in ancient greece. It's more or less well known that the irrationals were treated with skepticism but not just them, the rationals weren't really considered full fledged numbers, rather they were the ratios of numbers. Back then, to say that the rationals were numbers was analogous to saying that the relationships between things are as real as the things themselves. Here is some recently recovered work by Archimedes that made use of infinitesimals to resolve some problems in integral calculus.

http://en.wikipedia.org/wiki/The_Method_of_Mechanical_Theorems

He titles it "The Method" because he didn't consider it rigorous mathematics (for making use of infinitesimals, which were also not considered legit numbers). It's more or less supposed to be just an explanation of his reasoning. At any rate, the techniques are pretty clever and you can kind of see the mindset that people had when doing geometry in that era.

People have come up with some ridiculous solutions.

13 move octagon.

this bug has been persitant.

Why is it not turning grey for the insider circle achievement. It is 17 moves but refuses to give me the achievement despite being inside the starting circle. You can see all inside 3 circles are connected with each other and the outside circle exactly.

>>6512195
I got my first one in 17. Will be trying to improve the design.

16 gon lol

>>6515388
its beautiful almost looks like electron orbitals or something.

>>6515292
a flaw in the matrix.

>>6515201
Yeah, this website inspired me to make an elementary cellular automaton, and everything on it deserves a medal. It's amazing.
This game, though. Thank you based nerd^3 for rewarding me with this nectar of the gods.

>>6515259
It's not exact.
To get it perfect, start from a square, i'll let you figure it out from there.

>>6515316
The circles aren't tangent to one another. Look at the intersections.
Two points of intersection between each.

Someone make a heptadecagon, I made one in geogebra but not in agg. You get a cookie if you do.

>can't get it under 9 moves
God fucking dammit

>>6515517
spoiler

>>6515521
Ooooh

I feel retarded though.

>>6515528
wait until you get to pentagon

>>6515388
Generalise for N-Gon.

I didn't mean to do this , I was playing around and it just came up.

Spoiler
Circle Pack 7 14 moves

Hey guys wat are all the possible relative length line segments that you can construct with this bitch?

>>6515620
Impossible

>>6515467
It is exact, I actually used almost the same construction for a square and just removed two sides (the thing that looks like a Z in the middle).

>>6515904
You can construct all the integers by using a circle of radius 1 over and over again along a straight line. Addition can then be done with unit circles on a straight line and subtraction works the same way just in the opposite direction.

Then you can do division and multiplication in the same way. Here's a picture for multiplication. Construct two straight lines with integers marked (as described above) so that they intersect only at 0. Then choose two points, p and q, from each line respectively. Draw the line from 1 to q, then draw another line parallel at p. It will intersect at pq, refer to the diagram.

(cont.)

>>6516366
Division works the same way, refer to the picture. Again choose points p and q and proceed as in the picture.

The reason the two methods above work is because they are similar triangles.

fuck you i did it

So close to winning! Just need 20 move dodeca and 12 move 4-circle. Any hints?

>>6516369

So far you'll note that we have a Field of rational numbers, closed under addition and multiplication.

The next technique is kind of a special case. You construct a straight line. Mark the unit length of 1 using your unit circle. Then proceed to mark some more unit lengths on your line. In this picture we've drawn 1 and then 4 more (the total length of our line is 5). We then find the center point of our line (use two large circles of the same radius at either edge, draw a line connecting their two points of intersection). At that center point we draw a large circle so that our line cuts it in half. Now we draw a perpendicular line (construction is the same as finding the midpoint) at the point 1. Circumscribe a triangle in our half-circle using the intersection of our new vertical line and the half-circle. I know this is confusing so just look at the picture as it will make things clearer.

Thanks to Thales Theorem we know that our triangle circumscribed in the half-circle is a right triangle. Furthermore we know that the two smaller triangles that make up the larger one are also right triangles. It's then somewhat easy to show that all of the triangles are similar by Angle-Angle-Angle (hint: each of the smaller triangles shares two angles with the larger one, but because the angles in a triangle must sum to 180 degrees then the third angle in each is determined). Now because they are similar their sides must be in ratio and we can determine the height of the vertical line. It will always be the square root of however many units we went to the right of the initial 1. In this picture it will be square root of 4. A theorem for the construction of square root of n (where n is constructable) easily follows.

There are a handful of other ways to construct square roots but this is the best, imo.

(cont.)

>>6516419

Being able to construct square roots we are now able to add field extensions to our field of rationals. You can proceed like in this picture (going downwards is just taking the square root repeatedly).

You cannot construct cube roots and a ton of other lengths though. I've never quite found the explanation satisfactory though because square roots as a whole all rely on the technique in the last post and it just kind of comes out of nowhere.

>>6516419
i remember it when the teacher proved that trisecting angle and duplication of the cube were impossible with compas and straightedge. It's pretty cool to see that problem was open for like, 2000 years and then the solution came from algebra and is so simple...

>>6515904
anon >>6516419 is showing you how to construct effectively the lenghts you want but finding a large number of the lenghts you can't have is pretty easy. (if you know what a field extension is)

The only things you can do to construct new points is intersecting lines and circle between them. if you right the equations of a line and a circle in general you see that solving the intersection of two line doesn't make you go "out" of the field you were working on. However the intersection between a line and a circle does that, you have to take a square root of something : you (possibly) have an extension and its degree is 2. For the intersection of two circles it's the same, you have a square root and so you(possibly) have an extension of degree 2.

So all you can have is successive extensions of degree 2. The impossibility of duplicating the cube after that is obvious : the third square root of two has a transcendental degree of 3 over the field of rationals. The same goes for trisecting most of the angles and squaring the circle.

>>6516446
>The only things you can do to construct new points is intersecting lines and circle between them. if you right the equations of a line and a circle in general you see that solving the intersection of two line doesn't make you go "out" of the field you were working on. However the intersection between a line and a circle does that, you have to take a square root of something : you (possibly) have an extension and its degree is 2. For the intersection of two circles it's the same, you have a square root and so you(possibly) have an extension of degree 2.

This is a much better explanation for why you can construct square roots but not other things. Thank you for putting it so clearly and succinctly.

>>6516452
i think there is a little missunderstanding here.

I showed what we possibly could construct : addition multiplication and square roots. we can't do more than that. But maybe we can't even do that ?
hopefully you showed (assuming you're >>6516430) that those things are constructible and how to do it ( addition multiplication square root ...).

Some questions still remains, i let you see for yourself :
https://en.wikipedia.org/wiki/Constructible_number
https://fr.wikipedia.org/wiki/Th%C3%A9or%C3%A8me_de_Wantzel (french)
https://fr.wikipedia.org/wiki/Th%C3%A9or%C3%A8me_de_Gauss-Wantzel (also french)

>>6516532
>Some questions still remains, i let you see for yourself :

but i warn you that this is getting more tricky

>>6516532
Yes, you're right that is me. I showed how to construct those numbers but I did not show that it's all we can possibly construct. I had seen results about the intersections of lines/lines, lines/circles, circles/circles before but for some reason I never connected it to this argument.

I had also seen some arguments that used complex numbers for why certain polygons couldn't be constructed. For example, you compute the nth roots of 1 on the complex plane (also called the complex roots of unity) and this produces the complex numbers corresponding to the lengths you're interested in constructing on your constructed euclidean plane. Then you just prove that those lengths can't be constructed and so neither can the polygon.

I have found this all very interesting but I've had trouble finding any rigorous references that aren't just a small footnote in an algebra book, pop-math books aimed at laymen, and an actual constructionist geometry book full of constructions and strictly geometric proofs.

>>6516366
>>6516369
>>6516419
>>6516430
>>6516446
Dude, you're the tits. Thanks for this.

I saw this yesterday and it's a good thread that deserves a bump.
Regards, not OP.

>>6515781
>>6516376
13 moves.

Where's your Shiva now, atheists?

>>6518498
>>6516376
mine is the 'within the origin circle'
impressive though

>>6510254
What is this.

>>6518570
WHAT. NO. WHY. What are these points doing here?

>>6518570
yeah... you don't want to zoom to much, trust me on that...

>>6511920
Yes but I'm making a circle.

>>6510254
>mfw I made this thread months ago
>mfw no one replied
I found the least moves for everything and then some tho so, whatever.

>>6519479
Taking requests, for I have mastered this art form.

>>6519479
It's okay.
I've had it bookmarked since you posted it months ago.

>>6519487
Thanks Anon, for I now see that my thread was not in vain.

>>6519479
I'm sorry, anon. I didn't mean to steal your thread.
I suppose I just got lucky!

>>6519479
How do I get least moves for dodeca and 4 circle? It's all I have left!
Just a hint, though!

>>6519572
Hmmm.... I'm not very good at hints, but I'll try.
For the dodeca, you will need to construct a hexagram in a seed of life pattern (look it up).
Sadly, I'm not sure how to explain the four circle without giving it away. Sorry.

BUMP

>>6519479
I'm not OP but I'm sorry for not seeing your original thread and bumping it like I did for this one.
It breaks my heart when good threads get drowned in shit.

Here, have my eye.

>>6519770
Fair enough, I'll forgive you. I guess.

give me the three fucking circle pack please.
i have eleven moves and need to get it down to 9. have no fucking idea how. this is getting irritating because i finished everything else. anything to bisect an angle takes me longer.

>>6519892
Click on the figure in the challenges panel to reconstruct it. Make a new layer and analyze your construction, find out what operations will give you the necessary critical points, and start with different circles in the construction to find out the simplest ways to get to those points given any two initial points.

>>6519921
Yeah, I do know about that but I cannot reduce it even by one move.
Dodecagon, Pentagon and Octagon minimal forms were easier than this to me for some darned reason.

Here it is in pic related. How do you think I could optimize this?

>>6519932
that one down and left from the center doesnt look like it needs to be there

>>6519966
>down and left
down and right

bumping due to the current barrage of shitposting on the front page

>>6510254


what am I doing wrong?

>>6520789
yoou need to have the triangle around the center of one of the original two dots and the circle to be the first circle you can make between this original points

>>6520794
ooooooh ok thanks!!

>>6519968
>>6519966
I'm not that guy but I'm pretty sure you need that in order to construct the radius for the first circles you make.

>>6520789
Your triangle is not inside the original circle