hyperboloid

The mathemagician’s room - by, now it comes, meeeee :P

It’s a place full of wonder, strangeness and bizarre beauty: the room - or should I say space - of a “mathemagician”…

It’s a common optical illusion, a kind of tricky paradox. “If our brains are simple enough for us to understand them, then we’d be so simple that we couldn’t.” ~ Quote by Ian Steward (a mathematician)

In my mind, mathematics equals (somehow) magic, mathematics is magic, therefor “mathemagic” :P

The floor self is an impossible figure - a typical symbol for paradoxes… 
The spirals on the chessboard-floor are Fibonacci spirals. (If you want to know more, then check out my Phi/Fibonacci tag.

This drawing is mostly about the beauty of Phi Φ, Psi ψ and the Fibonacci-sequence.

The fibonacci sequence
The fibonacci sequence works this way:
1+1 = 2
1+2 = 3
2+3 = 5
3+5 = 8
5+8 = 13
8+13 = 21
13+21 = 34
21+34 = 55
34+55 = 89
55+89…
… and so on…
The sequence then is: 1;1;2;3;5;8;13;21;34;55;89;…….

Let’s play with these numbers!
1/1=1
2/1=2
3/2=1.5
5/3=1.66666…
8/5=1.6
13/8=1.625
21/13=1.615384 …
34/21=1.6190476…


…If we continue this more and more, it will come closer and closer to a mathematial constant: 

Phi Φ
Btw: 
Φ=1.618033988749894848…
It can be calculated as followed:
Φ= 1+√5’ /2
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Phi, the golden ratio as ratio:
If you have two lines:
The proportions are 1to1.618:
Or here demonstrated: 10 to 16.18
Like:  _ _ _ _ _ _ _ _ _ _ (10) _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ (16 (aka~16.18)lines) This is the proportion/ratio stupidly demonstrated in a text. xD


Psi ψ
Psi is the golden angle. Its value is about 137.5°.
So.. Let’s connect Φ with π (Pi)

ψ = 2π -2π/Φ = ~2.40 And now we show this as an angle:
2.40/2π = x/360°
(2π is the circumference of a circle (consider without r or r=1 :P ); 2π is equal to a 360° circle: So 2.40 of 2π (It’s about 6.2831) is equal to x of 137.5°)
And here the x equals ~137.5° :P

2.40/2π = x/360° : And now we solve the equation for x → *360
x = 2.40/2π *360 = 137.5

Yepp, that’s math :P

The golden angle can be found in nature almost everywhere!
Did you ever wondered about the awesome arrangement of the blossoms’ leaves? 
Check out this picture:

 

I would also recommend you to read the golden angle article of wikipedia

Math can be really beautiful - Psi, Phi and Fibonacci show you the most awesome aspects of math! :)
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After all I would really recommend you to visit my phi/fibonacci tag again^^
There you can find REALLY awesome things, such as finding Phi, the golden ratio ind the proportions of your body, your hand, bee population, plants etc…
it’s just WOW! :)

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This picture is dedicated to the beauty of mathematics, as well as the majesty of the Fibonacci-sequence, the golden ratio and the golden angle, which appear to be a sort of important algorithm in life… :)

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Because I am a real math advocate, here are some quotes too:
• “Mathematical beauty is a highly abstract, inner beauty, a beauty of abstract form and logical structure, a beauty that can be observed, and appreciated, only by those sufficiently well trained in the discipline.” ~ Keith Devlin
• “The difference between the poet and the mathematician is that the poet tries to get his head into the heavens while the mathematician tries to get the heavens into his head.” ~ G.K. Chesterson
• “Nevertheless, the fact is that there is nothing as dreamy and poetic, nothing as radical, subversive, and psychedelic, as mathematics. It is every bit as mind blowing as cosmology or physics… and allows more freedom of expression than poetry, art, or music…
Mathematics is the purest of the arts, as well as the most misunderstood." ~ Paul Lockhart

Have a good day :)

Parabola? Half a hyperbola? What is that curve?

Submitted by Djohariyea:

I saw this gif floating around on my dashboard

It’s a straight bar skewed at what looks like a 45 degree angle? As it rotates through the vertical plane, it traces that curve. I was wondering if that curve has is a piece of a certain conic section? It kinda looks like a hyperbola to me, but I was wondering if you had any insights based on how the mechanism works, if that’s a clue to what that curve is…
Thanks!

Hi, Djohariyea! Firstly, I’d like to apologize for waiting so long to respond to your submission. When I still had school, I simply did not have time to deal with my inbox, but now it’s summer! I know your question could be answered with one sentence, but I’m going to a full-blown explanation for followers who may want it.

Alright, this is stuff you need to know to understand what’s going on:

  • Hyperbola  (it’s a mathematical curve, one of the conic sections)
  • Hyperboloid of one sheet (it’s a three-dimensional shape obtained by revolving a hyperbola around its semi-minor axis. It is a ruled surface.)
  • Ruled Surface (a surface where for every point on that surface, there is a straight line that lies on that surface).

What’s going on?

Look at the straight bar in the gif. More importantly, look at the 3D shape the bar seems to be tracing out in the air as it spins. Hopefully, you’ll notice that it’s making this sort of shape: 

It’s a hyperboloid. A hyperboloid is a ruled surface, meaning that it’s curved shape can be created using straight lines. Have you ever heard of a mathematical envelope, or string art? Same idea. Here is a picture of a hyperboloid with it’s ruled surface much more apparent:

In this hyperboloid I’ve just showed you, the curved, hour-glass-like shape is made completely out of straight lines (string!). (This particular hyperboloid is actually double-ruled because there are lines going both up and down, criss-crossing.) But what do hyperboloids and ruled surfaces have to do with the gif? Look at the gif right now and imagine that the bar leaves a string behind, suspended in the air, at every single position that it is in during one rotation. It’s going to look a lot like the string hyperboloid above. What I am trying to say is that the tilted bar’s motion, as it revolves, takes the place of all the straight lines that make up a hyperbola. In other words, one moving straight line can act as many. This is what I mean, and this is what is going on in the submitted gif: 

So, to finally address you original question, what the heck is that curve that the straight bar passes through? I think it is this:

And what is the name of the curve? Well, you can obtain a hyperboloid by rotating a hyperbola around its semi-minor axis, so I believe that the curve is a hyperbola. Here’s a gif illustrating this idea: 

The End. Thanks for the submission! 

Sources of all pictures in order of appearance, disregarding the submission: Hyperboloid 1, 2, 3, 4, and 5.

TL;DR: The bar in the gif makes a hyperboloid and the curve the bar passes through is a hyperbola.

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