yay, my site made it to AMS blog. 〔Astroid as Catacaustic of Deltoid By John Baez. @ blogs.ams.org…〕
it's written by the redoubtable mathematician John Baez. Baez is great, in that he writes serious math for any math undergraduate to appreciate, as opposed to many math popularizing authors who write for the laymen.
i did my curves project Visual Dictionary of Special Plane Curves mostly in 1994 〜 1997, almost 2 decades ago, while i was a college student. I never seen the proof of how Deltoid's Catacaustic is a Astroid. I recall trying to, but it was too difficult for me back then. I haven't done much math since.
Do you know a proof of how Deltoid's Catacaustic is a Astroid? Post to John's g+ post. Thanks.
Equiangular Spiral, also known as log spiral, has the property that the angle of tangent to center is constant.
Mathematician Jacob Bernoulli (1654 〜 1705) requested this spiral be engraved on his tombstone with the epitaph:
In many fields of mathematics, morphism refers to a structure-preserving mapping from one mathematical structure to another. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms are functions; in linear algebra, linear transformations; in group theory, group homomorphisms; in topology, continuous functions, and so on.
In category theory, morphism is a broadly similar idea, but somewhat more abstract: the mathematical objects involved need not be sets, and the relationship between them may be something more general than a map.
The study of morphisms and of the structures (called objects) over which they are defined, is central to category theory. Much of the terminology of morphisms, as well as the intuition underlying them, comes from concrete categories, where the objects are simply sets with some additional structure, and morphisms are structure-preserving functions. In category theory, morphisms are sometimes also called arrows.
In topology, two continuous functions from one topological space to another are called homotopic if one can be “continuously deformed” into the other, such a deformation being called a homotopy between the two functions.
is there a homotopy that maps identity in the complex plane to 1/conjugate[z]? or sin[z]?
i think there is. It's obvious, that if the 2 spaces are topologically equivalent, there always is, the question is to find the homotopy. In my case, just use the idea of
Geometric Inversion. Let p be the point in domain and p' in range, then just smoothly swap them by gradually narrowing their distance.
one of the most fruitful thing g+ has ever done for me since its beginning is discovery of
He's a mathematician, and also a well-known writer (even before blog days) He writes a lot, but even when writing research level math, he made it easy for undergrad to understand. And, takes the time to write the interesting aspect, and answer and discuss with your comments/questions. (thus, comments on his post/blog are often very high quality as well) I see that he also sometimes write non-math related things, that touches on history, art, linguistics, all in a very appetizing way with quality/rare photos (and yet not the trite, mundane, beaten-horse types you find daily from social networks). Incredible!
you can read his bio on wikipedia and also follow links to his blogs.
i'm learning lots stuff from John C B. Lots thoughts hard to summarize nicely.
For one thing, related to SEO, is that it solidifies the idea that in order to get more readers, one should really focus on readers — so-called “engagement”. For example, say, instead of writing 4 posts per day, write just 1 and put the time of the 3 into that 1, to include quality image/illustration, answer questions, iron-out hand-waving. In other worlds, this is really the road for professional blogger. (you might not want to have lots readers, or shudder from the idea of wanting to be “popular”. But if you write publicly, more readers is positive in psychological and practical and philosophical ways. “Readership” defines “authorship”.)
JCB is also pulling me back into math. Such a black hole of pure beauty. The depth of which tantamount the very question of existence and universe.
JCB also sets a good example of doing good in a solid way. (as opposed to the countless shallow and crowd-pleasing blogs, exemplified by the marketing droids of Google of recent years (⁖ Google Science, Google Doodle, Google pro-lgbt, …), and countless fanatical “left-leaning liberal” American slackavitists daily pushing their selfish-opinions in the name of greater good.)
GeoGebra was open source (GPL) for about 10 years (up to version 4.0), but since version 4.2, now only for non-commercial use. This is bate ＆ switch, but the problem is really open source. When it gets big, it needs funding, but nobody wants to pay.
Here's quote from Wikipedia GeoGebra on its licensing:
Most parts of the GeoGebra program are licensed under GPL, making them free software. However some parts, including the Windows and Mac installers, have a license which forbids commercial use and are therefore not free software. In practice, this means that non-commercial use by teachers and students is always free of charge, while commercial users may need to pay license fees. For details see the GeoGebra license description.
Since July 2010 the Debian GNU/Linux distribution offers a free version of GeoGebra 4.0 in which all un-free parts of the program were removed or replaced by free software. This version may be used for commercial purposes without paying licensing fees. However, starting with version 4.2 since December 2012, the license is changed to be more restrictive so that GeoGebra cannot be included in Debian GNU/Linux any longer. On the other hand, the software can still be downloaded from its official download page free of charge for many platforms (including Debian as well).
GeoGebra is a Java Applet. But since Apple Apple killed Flash in 2010, as well as not including Java, Java applet is pretty much dead (it doesn't run on any Apple iOS nor Google Android phone/tablet).
A great blog from a architect. Always lots of most beautiful math images. Be sure to checkout his past articles. 〔eat-a-bug: Art, Design, Architecture, ＆ Technology By Lorenz Lachauer. @ eat-a-bug.blogspot.com…〕