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Malin Christersson
Lived in Lund
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Malin Christersson

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The animations are beautiful. Apparently they are made by the same guy (Dan Gries) that made the digital paper snowflake --at http://rectangleworld.com/blog/
 
Planar Choreographies

Analysing the paths of celestial bodies under gravity is notoriously difficult for anything more complicated than just two bodies such as the Moon around the Earth. Over the centuries it has been necessary to rely on simplifying assumptions and the mathematics of numerical approximations to come close to predicting the possible paths of bodies with various masses, starting points, and velocites in a gravitational field.

Amazingly, for the purposes of the original classical challenge, only the two-body problem and a restricted 3-body problem have been solved! In the last twenty years, however, with the help of computers, a special class of pretty but abstract gravitational n-body solutions called Choreographies has been discovered. In a Choreography all the bodies have equal mass and follow each other evenly around a closed but possibly intersecting path. Rather like police display motorcyclists whose paths intersect at speed but who don't crash into each other, to qualify as a Choreography, the bodies are non-colliding.

Retrospectively, Lagrange's famous discovery in 1772, where three bodies are at the corners of a rotating equilateral triangle, is a Planar Choreography. Planar Choreographies are those restricted to two dimensions and +Katie Steckles and +James Montaldi have amongst other things categorized them for us in the paper below and then asked +Dan Gries to make them visible to us with these lovely

Interactive Animations: http://goo.gl/KkO7UG

The principal aim of the present paper is to make systematic the combination of topological (braid) methods and symmetry methods. We begin by classifying all possible symmetry groups arising for (collision-free) choreographies in the plane, and then proceed to study symmetries in loop space, firstly in general and then referring specifically to choreographies. The work is an extension of the work presented in the second author’s thesis

Paper (open): http://goo.gl/I4Kt6t

n-body problem (Wikip): https://goo.gl/MC6g9E
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Lovely picture.
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Malin Christersson

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Open Access
 
Elsevier news

Hard to believe I know, but a lot of people are getting cross with Elsevier again. Elsevier have recently revised their policy about sharing of articles, which can be examined in a web page written by Alicia Wise that goes by the superbly ironic title, "Unleashing the power of academic sharing." (I think the right word for a title like that has to be "wisecrack".) It's here:

  http://www.elsevier.com/connect/elsevier-updates-its-policies-perspectives-and-services-on-article-sharing

One of the highlights is that after publication a subscription article can be shared "As a link anywhere at any time." This policy is explained on another Elsevier page as follows: "If you are an author, please share a link to your article rather than the full-text. Millions of researchers have access to the formal publications on ScienceDirect, and so links will help your users to find, access, cite, and use the best available version." (The relevant page is this one: https://www.elsevier.com/about/policies/article-posting-policy#published-journal-article.)

I actually don't find Elsevier's policies all that unreasonable. For one thing, they are pretty liberal about preprints in mathematics, allowing you to post to the arXiv the version that takes into account comments by the referees. However, it is notable that they don't allow just any old repository, so they are clearly making special cases for certain subjects, while making sure that freely available preprints don't become the norm in the big-money subjects like biology and medicine. But even that is reasonable if you look at things from Elsevier's point of view: they have a business model that would be seriously threatened if you could get the information you wanted without subscribing to their journals. But their critics are also reasonable from their point of view: it is not good to have large amounts of the scientific literature behind paywalls. Also, Elsevier's defence of its policy -- that Science Direct is a wonderful resource that helps you find the best version of the article -- is ludicrous.

Basically, as everyone knows, we are seeing a clash between the future and the past. The future is surely destined to win eventually, and the main question is how much money the past can rake in before it does.

On another topic, I missed this when it happened, but on the 12th of May the Cost of Knowledge boycott passed the 15,000 mark. It now stands at 15039.
Statement against Elsevier's sharing policy. Organizations around the world denounce Elsevier's new policy that impedes open access and sharing. On April 30, 2015, Elsevier announced a new sharing and hosting policy for Elsevier journal articles. This policy represents a significant obstacle to ...
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This is an interesting analysis. The analysis is not about security keys but about people, and as  +Jürgen Christoffel puts it:  about "bureaucratisation of the Internet".
 
"""Much like the patent system and the Secure Boot system, an HTTPS requirement means that you have to check with a bureaucrat before you post code you wrote to the world. A kid who wants to ditch WordPress and make up his or her modern and hopefully more fun incarnation of the Caltech Divinity School now has more hurdles in the way."""

#NewWeb
Mozilla, the foundation that maintains Firefox, has announced that it will effectively deprecate the insecure HTTP proto…
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+Nico Gerrits I agree, the comments are interesting as well.
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+American Patriot I suppose I live in some sort of filter bubble in which the threats of a surveillance society is an important issue, and outside that bubble most people don't care. Which is a rather depressing thought.
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Beautiful!
 
algebra + droste 2d + fraktal + Hyperbolic Tilings
Malin Christersson: Hyperbolic Tilings http://www.malinc.se/m/ImageTiling.php
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Thank you for your kind words +Liang Zhao
This is a remake of an old program made in GeoGebra+Python, then remade to Java, then remade to Javascript. The main object is a Javascript implementation of a generalised circle (https://en.wikipedia.org/wiki/Generalised_circle) with appropriate methods for reflection in line and circle inversion. The hyperbolic methods are basically Javascript implementations of  the GeoGebra constructions at http://www.malinc.se/math/noneuclidean/poincarediscen.php . The tiling itself is implemented as a tree.  
The program is a hideous mix of object oriented and functional programming, since I don't know (or like) Javascript, and it’s optimised for speed rather than readability. 
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Hyperbolic Tilings of Images

I have remade an old program for making hyperbolic tilings to a Javascript version. An image is cropped to a hyperbolic polygon in the center of the disc. That polygon is then reflected repeatedly to fill out the tiling. 

When an image of a face is reflected it gets distorted (from an Euclidean point of view). Depending on the tiling, the distortions range from “slightly strange” to grotesque. The tiling of Poincaré is “slightly strange” (I hope) while the hyperbolic self-portrait is creepy (at least according to my kids).

Make Hyperbolic Tilings of Images: http://www.malinc.se/m/ImageTiling.php
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Cool Thanks! :)
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It's in Spanish, with English subtitles.
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+Michelle Beissel I agree, I smiled through the entire presentation. :)
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+Paula Morais That's true, to make it e-learning. :)
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A bottle opener would be nice.
 
Hyperbolic Buttons
     
+Malin Christersson has made a Javascript version of her hyperbolic tilings tool, which means that you can now simply drop your picture on her web page and make your own hyperbolic art – from which you could make a button, bottle opener or beermat!
     
Here: https://goo.gl/yCqdhh

Buttons: http://goo.gl/ZNzTdd
(example only – there exist other services)

There is a relationship between certain fractals and these hyperbolic tilings.

Few people know, however, that fractal pictures like this one are intimately related to tilings of what mathematicians call hyperbolic space. One such tiling is shown in figure 1a below. In contrast, figure 1b shows a tiling of the ordinary flat plane. In this article, which first appeared in the Proceedings of the Bridges conference held in London in 2006, we will explore the maths behind these tilings and how they give rise to beautiful fractal images.

More here (see animations): https://goo.gl/11VLpo

Artistic Hyperbolic Geometry Tutorial: http://goo.gl/e0jPBY   

From...

Math and the Art of M. C. Escher: http://goo.gl/EZzLRz

Snowflake wallpapers: http://goo.gl/S2P6QD
Courtesy of +Alexey Kljatov.
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It's everywhere.
:D
 
A few more instances of the golden ratio and the golden spiral in nature, technology, and the arts, almost as convincing as the usual suspects.
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I thought xkcd 1488 brought us to the conclusion that this is total BS. [ https://xkcd.com/1488/ ]
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Mapping pixels from a Euclidean to a hyperbolic polygon

(follow up on my previous post http://goo.gl/9kEWOz )

In my original hyperbolic tiling of images, an image is cropped to a hyperbolic polygon that is then reflected repeatedly. I got a suggestion from +Kerwin Vincent  to distort the first image as well, and distorting all tiles is a better idea than treating one of the tiles like some sort of Euclidean alien in a hyperbolic universe. Distorting the first tile takes many extra calculations, and the result is only visually noticeable at the edge of the polygon, but as a result: All tiles are equal (but one tile is more equal than the others)!

Make Hyperbolic Tilings of Images: 
http://www.malinc.se/m/ImageTiling.php
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I got the scaling right. It's a rather cool effect. 
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The art of making programming errors

Some programming errors are more appealing than others.
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+Roice Nelson wow! You really could decorate your living room with beautiful computer bugs!
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In her circles
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Lund
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mathematics - computer science - digital freedom
Introduction
My posts are mainly about mathematics and programming, from an educational point of view. I am currently pursuing a PhD in educational research. My research is about kids learning how to program, and the math included when doing so.
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