# 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:

External image

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! :)

External image

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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 :)

8

I made these models of curved surfaces: z=x*y, z=x^2+y^2, x^2+y^2-(¾)z^2=¼ and x^2+4y^2-(¾)z^2=¼.

It was hard to make;-)

I think no surfaces present which contain straight lines in other than hyperbolic paraboloid and one sheet of hyperboloid. Is it right?

The gif animations of these curved surfaces:

Hyperboloid Night

Hyperboloid.

3

Hyperbolic paraboloid and One sheet of hyperboloid. ;-)

The models of these curved surfaces:

Hi! I was having a random thought on the Sagrada Familia "controversy": I think they did the right thing leaving it unfinished for such a long time. They gave people the chance to experience architecture in a very peculiar way, the unfinished building became a unique work of art. Now though I am also very happy they decided to finished it. It's like a bonus: two work of art in one! What are your thoughts on the matter?

There was something undeniably romantic about the idea of Sagrada Familia as an unfinished work. Gaudi was someone ahead of his time that designed his building in paper and kept designing during construction exploring elements taken by nature. Forms known today as  helicoids, hyperboloids and hyperbolic paraboloids are incorporated into the design in ways that is difficult to understand without the help of computers.

Via

Line and Circle (Rod and Ring) to Hyperboloid and Sphere, three photogrpahs from an experiment, class by Joost Schmidt, construction: Heinz Loew / photo: Edmund Collein, around 1930.

Bauhaus-Archiv Berlin / © Ursula Kirsten-Collein / Irmgard Loew

Hyperboloid Night. Rejected.

Tell Me What You See #007 Mathematical Concepts Part_02

Visualizing sine (red) on the Y axis and cosine (blue) on the X axis. The relative position of the circle is shown in black:

This shows the same thing, but a bit more simply:

Here’s how sine and cosine apply to triangles:

Cosine is the derivative of sine:

Tangent lines:

Flipped on its side, the shape begins to make more sense:

Converting a function from Cartesian to Polar coordinates:

Drawing a parabola:

The Riemann sum is the approximate area under a curve:

Hyperbola:

Translating that into 3D, you get a hyperboloid. Believe it or not, it’s made with completely straight lines:

Seriously. You can even make it do this:

## Here are two more examples of straight lines tracing hyperboloids.

10

Byriah Loper (from top):

K5: Twenty Interlocking Tetrahedra, version 2

Five Interlocking Irregular Hyperboloidal Rhombic Dodecahedra; Event Horizon: Twenty Interlocking Irregular Augmented Tetrahedra

Five Interlocking Irregular Hyperboloidal Dodecaugmented Cuboctahedra

Ten Interlocking Triaugmented Equatorially Diminished Triangular Bifrusta; Ten Interlocking Irregular Hyperboloidal Triaugmented Omnitruncated Digonal Dihedra

Five Interlocking Irregular Hyperboloidal Truncated Triakis Tetrahedra

Fifteen Interlocking Wrinkled Rectangles; Five Interlocking Irregular Hyperboloidal Hexeaugmented Truncated Tetrahedrically Distorted Hexahedra

Ten Interlocking Triangular Prisms #4

2

Hyperboloid of one sheet.

Hyperbolic Paraboloid
A concept model I produced during experimenting with different tower forms.

finished!

hyperboloid structures

inside a hyperboloid cooling tower

Layout.