We have three colored segment in this animation. Surprisingly the length of the longest one is always the sum of the length of the two smaller ones.
This is actually a very special case of Ptolemy’s theorem. The theorem gives a connection between the sides and the diagonals of a cyclic quadrilateral. In this case the length of the dashed lines is equal so the theorem can be simplified to the statement above.
1. Take a circle and draw some points on the boundary. For every point you draw, you must also draw its antipode (point on the opposite side of the circle).
2. Draw some points in the interior wherever you want.
3. Label the points either +1, -1, +2, or -2 as you wish. The only stipulation is that antipodes must have opposite sign.
4. Draw triangles however you want without crossing lines.
Tucker’s Lemma says that you will ALWAYS end up with at least one line that has endpoints of either +1 and -1 or +2 and -2. Try it! More info and proof here.
The answer is NO, you can not. This is why all map projections are innacurate and distorted, requiring some form of compromise between how accurate the angles, distances and areas in a globe are represented.
This is all due to Gauss’s Theorema Egregium, which dictates that you can only bend surfaces without distortion/stretching if you don’t change their Gaussian curvature.
The Gaussian curvature is an intrinsic and important property of a surface. Planes, cylinders and cones all have zero Gaussian curvature, and this is why you can make a tube or a party hat out of a flat piece of paper. A sphere has a positive Gaussian curvature, and a saddle shape has a negative one, so you cannot make those starting out with something flat.
If you like pizza then you are probably intimately familiar with this theorem. That universal trick of bending a pizza slice so it stiffens up is a direct result of the theorem, as the bend forces the other direction to stay flat as to maintain zero Gaussian curvature on the slice. Here’s a Numberphile video explaining it in more detail.
However, there are several ways to approximate a sphere as a collection of shapes you can flatten. For instance, you can project the surface of the sphere onto an icosahedron, a solid with 20 equal triangular faces, giving you what it is called the Dymaxion projection.
The Dymaxion map projection.
The problem with this technique is that you still have a sphere approximated by flat shapes, and not curved ones.
One of the earliest proofs of the surface area of the sphere (4πr2) came from the great Greek mathematician Archimedes. He realized that he could approximate the surface of the sphere arbitrarily close by stacks of truncated cones. The animation below shows this construction.
The great thing about cones is that not only they are curved surfaces, they also have zero curvature! This means we can flatten each of those conical strips onto a flat sheet of paper, which will then be a good approximation of a sphere.
So what does this flattened sphere approximated by conical strips look like? Check the image below.
But this is not the only way to distribute the strips. We could also align them by a corner, like this:
All of this is not exactly new, of course, but I never saw anyone assembling one of these. I wanted to try it out with paper, and that photo above is the result.
It’s really hard to put together and it doesn’t hold itself up too well, but it’s a nice little reminder that math works after all!
“In my remembrance of the compassionate (rahma) spirit, and as a woman of nomadic heritage, I try to reclaim the womb (rahim) and my connectedness to a nourishing power that is always with me wheresoever I wander. Water and its circumbulative connection to the Divine breath are primordial symbols inherent to the vitality of my journey.”—Artist Amina Ahmed.
A 75 × 112 rectangle can be cut up into 13 squares, which can be rearranged into the initial rectangle in two different ways. One way was found by Brooks, who was so pleased with this dissection that he made a jigsaw puzzle of it, each piece being a square. His mother then tackled the puzzle and eventually succeeded in putting the pieces together, but not in the way Brooks has found!
“The facts that musical notes are due to regular air-pulses, and that the pitch of the note depends on the frequency with which these pulses succeed each other, are too well known to require any extended notice. But although these phenomena and their laws have been known for a very long time, Chladni, late in the last century, was the first who discovered that there was a connection between sound and form.”
The Mandelbrot Set Drawn in a Microsoft Excel 2010 Spreadsheet
I have a lot of extra time on my hands as an IT intern, so I decided to learn Visual Basic for Applications (VBA). After making a couple of small applications using VBA for Excel, I decided to attempt something more entertaining: the Mandelbrot set! For those unfamiliar with the set, I really recommend checkingitout!
The Mindful Programmer (Joni Salonin) wrote an awesome blog post about creating a Mandelbrot set drawer in a C-family language. Using it as a guide, I drew a set in black and white, then grey-scale, then eventually color. Instead of having the program draw to individual pixels, I made the cells of the spreadsheet super tiny
squares, zoomed out to 10%, then had the program set the fill color of each individual cell in accordance with the Mandelbrot set equation. As you can probably imagine, it takes quite a while to render, especially when the number of iterations is set high. For these sets, I set the number of iterations to 30 — it takes around 3-6 minutes to render.
Bonus pic - Mandelbrot set in grey-scale:
Explanation of pictures at the top: The top picture is a full-screen snapshot of my full-color Mandelbrot set. I really wanted it to look like the Wikipedia page’s set so I had to be careful with my color/gradient choices! The bottom two pictures show a spreadsheet selection of 21 by 22 cells first at 10% zoom, and then at 214% zoom. Imgur album. Code (I apologize for its messiness).
Check out the full 240 page book cataloguing this years festivities and exhibitions here! They had some nice things to say in highlighting the GIF “Dust Loops” below:
FILE GIF 2016 – patterns GIF as time paradox: between transience and permanence
Patterns are repetitions that occur in time. There is no realization of a pattern without the temporal element that makes possible the perception of something that repeats. Time, on the other hand, is unfathomable. We develop strategies to perceive it, demarcating seconds, minutes, hours and years. We think we can preserve it somehow, such as when we use photography to recall a moment that is gone. However, the reality is that time escapes us all the time, at every breath.
The perception of time has been problematized by many theorists that study the impact of technical and technological inventions in our everyday life since the end of the 19th century with the Industrial Revolution. With the advent of the telephone and television broadcasts, the exchange of information starts taking place “live”. If before we would take days, weeks or months to receive some information, now, with internet and mobile devices, this information becomes available almost at the same time that it is produced for geographically distant locations. Time and even space become ubiquitous.
GIFs are a typical phenomenon of the ubiquitous online networks in the early 21st century. Invented in the late 1980s, they were a way to produce, transmit and store images in low resolution compatible with the technology of the time. Today, this objective was diluted and they represent the dissemination of patterns that, precisely because of its scope, communicate with much of the web users, becoming memes, cultural phenomena whose information is propagated through the internet with great impact, viralizing. As soon as they are produced and disseminated, these memes are lost because of their ephemeral nature, but do not cease to exist. Thus, like memes, we can say that GIFs are paradoxical. Ephemeral and permanent, that is, related to patterns, therefore, to something that is formed on time, even if that time is short and ubiquitous.
GIFs and memes are generally produced by anonymous, but, increasingly, there are artists that choose this way to develop their poetic. In this edition of FILE, we present a selection of more than 40 proposals that emphasize the own medium structure addressing patterns, emphasizing or breaking them, whether they are behavioral (individual and social) or formal.
The highlight of the exhibition is “Dust Loops”, by Sumit Sijher. As the name suggests, the GIF contains points that move along the frame as dust particles. The movements seem chaotic and random, but the sequence has only 20 frames that repeat in 2 seconds. This makes curious the fact that we cannot follow the complete trajectory of a particle over these 2 seconds, not even over 10 or 60. According to the artist, we would need half an hour for this. This GIF meets the proposal of this exhibition, for time here is fundamental for its comprehension. It is a way to expand the notion of GIF, whose visualization usually is very brief, as well as its creative possibilities.
The production of Sijher and of all the other artists in the exhibition can be found online, spread through the internet in several sharing websites and platforms. They represent part of the diversity of GIFs produced in recent years, which result from different creative processes with algorithms, photographs, drawings and animations, but have in common the repetition that occurs in the physical and online networks.
- Fernanda Albuquerque de Almeida FILE GIF 2016 Curator
For reference sake, you can find the particular intothecontinuum GIFs and their abstracts that appeared in FILE, and the corresponding Mathematica code on this Tumblr blog here:
“Dust Loops” “The seemingly random and chaotic motion of 5,000 particles are captured in just 20 frames which repeat in a mere 2 second long loop, but results in a much longer apparent motion. If the motion of any single particle is followed it takes about 30 minutes for the particle to return to its original location.”
“A stack of blocks spin around at varying speeds: relative to the 1st block at the bottom of the stack, the 2nd block spins twice as fast, the 3rd block spins three times as fast, and so on… The overall motion results in fluctuating moments of order and disorder in the overall structure as it continuously twists about itself.”
“Perpetual Bloom”
“Using algorithmic randomness, various colored disks fade in and out while conforming to a 13-fold rotational symmetry giving the appearance of a kind of flower in a state of perpetual bloom.”
“A series of bell-like curves sway back and forth while being overlayed vertically, and having their motion slightly out of phase from the preceding layer. The resulting dynamics reveal apparent motion in both directions along the vertical; however, as made evident by the static horizontal lines, nothing is actually moving up or down in the visual field.“
