The Electronical Rattle Bag

Internet scrapbook of Paul Greer (@burningfp). Animation, etc.

Items tagged: #math

intothecontinuum:

In last weeks post, we saw how the motion of particles that move along straight lines creates the illusion of a spinning circle. This time we actually let the individual particles move in circular paths and observe various patterns that result when the relative phase of each particle is varied. Here, “phase” just means where along the circular path a certain particle is when compared to the others.

In the first animation, each of the particles arrive at the edge of the black circle at the same time to create the effect of a spinning and contracting/expanding circle.

In the second animation, the particles are phased just right to create the illusion of a circle that slides along the edge of the black circle. This is similar to the Tusi motion from the previous post except in this instance the circle doesn’t spin.

In the third animation, the phases are adjusted to make it seem like the particles move along a straight line that spins around, but really each particle is still only moving along a circular path. This is a somewhat opposite effect from the Tusi motion where the particles were always moving along straight lines.

Inspired by the not-Tusi-couple.

Mathematica code:


Manipulate[
Graphics[
{{Black,
Disk[{0, 0}, 1.05]},
Table[
Rotate[
{White, Opacity[o],
Circle[{.525, 0}, .525]},
n*2 Pi/m, {0, 0}],
{n, 1, m, 1}],
Table[
Rotate[
{White,
Disk[
.525 {1 + Cos[-2 Pi (p*n/m + t)], Sin[-2 Pi (p*n/m + t)]}, .02]},
n*2 Pi/m, {0, 0}],
{n, 1, m, 1}]},
PlotRange -> 1.1, ImageSize -> 500],
{{m, 8, "circles"}, 1, 20, 1},
{{o, .5, "path opacity"}, 1, 0},
{{p, 0, "phase"}, 0, 2, 1},
{t, 0, 1}]

theantidote:

Picture 1: wall mosaic on Darb-E Imam shrine (left) / atomic model of silver-aluminum quasicrystal (right).

Picture 2: infographic from the Nobel Foundation.

This year’s Nobel Prize in chemistry was awarded to Dan Shechtman for his discovery of quasicrystals. Quasicrystals, unlike traditional crystals, are aperiodic on the atomic level. Basically, their patterns don’t repeat. When Shechtman first saw this in an experiment in 1982, this was scientific heresy. Crystals were periodic, period. Shechtman must have made a mistake. But he hadn’t, and rather than sitting around sulking about his doubtful colleagues, he worked hard to eliminate possible errors and build further evidence for the existence of quasicrystals. The tide of evidence turned in his favor, and the field of crystallography was changed forever.

In hindsight, quasicrystals are the sort of thing that seem to be too beautiful not to exist. (Which is not to say that, just because a theoretical structure is beautiful, it always turns out to exist—it doesn’t.) Although it took until 1982 to find evidence of atomic patterns that were not periodic, aperiodic tilings show up on the walls of mosques as early as the 12th century.

(via proofmathisbeautiful: / science:)

blazinuzumaki:

sisterspock:

14-billion-years-later:

The Logarithmic Spiral

Now all you guys who are like “Yeah man the Fibonacci spiral is awesome” can just take a back seat here, because here we have the coolest of all spirals: the logarithmic spiral. Truth be told just about every time you’ve heard someone talk about the Fibonacci (or more accurately known Golden Spiral) they’ve been talking about this guy and just not realized it. The logarithmic spiral is given by the equation r=ae^(bθ) where r is the radius, a & b are positive constants and θ is the angle around the origin.

The logarithmic spiral also pops up quite often in nature, being the mathematical pattern behind such things as nautilus shells, Romanesco broccoli, spiral galaxies, the Mandelbrot set, storms, ferns and even sea horses.

Uzuumaaakiiii!
See also.

“Numbers it is. All music when you come to think. Two multiplied by two divided by half is twice one. Vibrations: chords those are. One plus two plus six is seven. Do anything you like with figures juggling. Always find out this equal to that. Symmetry under a cemetery wall. He doesn’t see my mourning. Callous: all for his own gut. Musemathematics. And you think you’re listening to the etherial. But suppose you said it like: Martha, seven times nine minus x is thirtyfive thousand. Fall quite flat. It’s on account of the sounds it is.”

– James Joyce from Ulysses (via theantidote)

“When I feel well and in a good humour, or when I am tak­ing a drive or walk­ing after a good meal, or in the night when I can­not sleep, thoughts crowd into my mind as eas­ily as you could wish. Whence and how do they come? I do not know and I have noth­ing to do with it. Those which please me I keep in my head and hum them; at least oth­ers have told me that I do so. Once I have my theme, another melody comes, link­ing itself with the first one, in accor­dance with the needs of the com­po­si­tion as a whole: the coun­ter­point, the part of each instru­ment and all the melodic frag­ments at last pro­duce the com­plete work. Then my soul is on fire with inspi­ra­tion. The work grows; I keep expand­ing it, con­ceiv­ing it more and more clearly until I have the entire com­po­si­tion fin­ished in my head though it may be long. Then my mind seizes it as a glance of my eye a beau­ti­ful pic­ture or a hand­some youth. It does not come to me suc­ces­sively, with var­i­ous parts worked out in detail, as they will later on, but in its entirety that my imag­i­na­tion lets me hear it.”

Psychology of Invention in the Mathematical World - Jacques Hadamard

via An Unquiet Mind

Mathematical habits of mind »

bobulate:

Ten mathematical habits of mind a teacher created for his sixth grade students:

1. Pattern Sniff
2. Experiment, Guess and Conjecture
3. Organize and Simplify
4. Describe
5. Tinker and Invent
6. Visualize
7. Strategize, Reason and Prove
8. Connect
9. Listen and Collaborate
10. Contextualize, Reflect and Persevere

Math lessons in everyday life. Everyday life lessons in math. Or as has been well said, “applicable to nearly anyone doing anything.” See also: Math Circles

ekstasis:

Galerie [DAM]Berlin - mlExhibitions

In the Process series Reas has explored the relationship between  naturally evolved and artificial systems. Organic forms are emerging  from precise mechanical instructions. The images visualise systems  moving or at rest. Reas is transferring his software pieces into  different media such as projection, images or sculptures. Each material  highlights a different aspect of the software.

Instructions for Process 13: Bisect a rectangular surface and define the dividing line as the origin  for a large group of Element 1. When each Element moves beyond the  surface, move its position back to the origin. Draw a line from the  centers of Elements that are touching. Set the value of the shortest  possible line to black and the longest to white, with varying grays  representing values in between.
(using  Element 1 from the Library: Form: Circle + Behaviour: Move in a  straight line + Constrain to surface + Change direction while touching  another Element + Move away from an overlapping Element)
Implemented by C.E.B. ReasFall 2009, Summer 2010Processing 1.0
“During  the last seven years, I have continuously refined the system of Forms,  Behaviors, Elements, and Processes. The phenomenon of emergence is the  core of the exploration and each artwork builds on previous works and  informs the next. The system is idiosyncratic and pseudoscientific,  containing references ranging from the history of mathematics to the  generation of artificial life.”
— C.E.B. Reas from the catalogue “Process Compendium 2004-2010”


(via Bruce Sterling)


yup

ekstasis:

Galerie [DAM]Berlin - mlExhibitions

In the Process series Reas has explored the relationship between naturally evolved and artificial systems. Organic forms are emerging from precise mechanical instructions. The images visualise systems moving or at rest. Reas is transferring his software pieces into different media such as projection, images or sculptures. Each material highlights a different aspect of the software.

Instructions for Process 13: Bisect a rectangular surface and define the dividing line as the origin for a large group of Element 1. When each Element moves beyond the surface, move its position back to the origin. Draw a line from the centers of Elements that are touching. Set the value of the shortest possible line to black and the longest to white, with varying grays representing values in between.

(using Element 1 from the Library: Form: Circle + Behaviour: Move in a straight line + Constrain to surface + Change direction while touching another Element + Move away from an overlapping Element)

Implemented by C.E.B. Reas
Fall 2009, Summer 2010
Processing 1.0

“During the last seven years, I have continuously refined the system of Forms, Behaviors, Elements, and Processes. The phenomenon of emergence is the core of the exploration and each artwork builds on previous works and informs the next. The system is idiosyncratic and pseudoscientific, containing references ranging from the history of mathematics to the generation of artificial life.”

— C.E.B. Reas from the catalogue “Process Compendium 2004-2010”

(via Bruce Sterling)

yup