Animation for MAT1300 assignment 4.

JBB: An Artblog!
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@quanticloop
Animation for MAT1300 assignment 4.
Nobody tells this to people who are beginners. I wish someone had told me. All of us who do creative work, we get into it because we have good taste. But there is this gap. For the first couple years you make stuff, it’s just not that good. It’s trying to be good, it has potential, but it’s not. But your taste, the thing that got you into the game, is still killer. And your taste is why your work disappoints you. A lot of people never get past this phase; they quit. Most people I know who do interesting, creative, work went through years of this. We know our work doesn’t have this special thing that we want it to have. We all go through this. And if you are just starting out or you are still in this phase, you gotta know that it’s normal and the important thing you can do is do a lot of work. Put yourself on a deadline so that every week you finish one piece. It’s only by going through a volume of work that you will close that gap, and your work will be as good as your ambitions. And I took longer to figure out how to do this than anyone I’ve ever met. It’s gonna take a while. It’s normal to take awhile. You just gotta fight your way through.
Ira Glass (via sumfiend)
Mark A. Reynolds - Greater and Lesser Dyad Series
Click on each image for details.
I’m bad at math, so the equation 2n+2n is 4n to me.
saying that “the diagramme commutes” is actually an abuse of notation.
Enduring Knots
For centuries, this intricately wrought interlace design—one of six similar motifs produced by Albrecht Dürer—has entranced and perplexed those who have encountered it. The intended purpose of the Knots, as Dürer referred to the series, as well as the precise date of their execution, remains unknown. Some scholars have postulated that the designs are patterns for embroidery or textiles. Others have suggested that they were ornamental designs for ceramics, or as labyrinthine didactic tools for training draftsman.[1] 1 What is certain is that the story of the Knots traverses culture and time, touching on the relationship of master and student, and the reuse and reinvention of a series of graceful, rational motifs.
Born in Nuremberg on the cusp of the German Renaissance, Dürer was a virtuosic artist and intellectual, renowned for his painting and draftsmanship as well as for his theoretical writings on proportion and geometry. One of his greatest legacies was his visionary contributions to the medium of printmaking. A master of both engraving and the woodcut, Dürer brought an unprecedented tonal and descriptive range to his prints, endearing his audience to printmaking and revolutionizing the potential of the medium. The decorative tradition of geometric interlace patterns predates Dürer by centuries, and was employed in Roman, medieval, and Islamic ornament. This remarkably complex woodcut, however, has its direct origins in the work of Dürer’s Italian contemporary, Leonardo da Vinci (1452-1519). All six interlace patterns, in fact, are copies of designs by Leonardo, for whom interlace motifs may have served as an informal, lyrical coat of arms. Vinci, his birthplace, translates as “to entwine”, and his notes reveal a preoccupation with the word gruppi, the Lombard term for “knots.” Interlace designs appear in many of Leonardo’s drawings and paintings, often as ornamental motifs worked into the hair or clothing of his subjects. The designs are also found in the decorative interlaced branches in the magnificent fresco ceiling for the Salle delle Asse at the Sforza Castle in Milan. To compose his circular knot designs, Leonardo would have constructed a grid of dots, weaving his drawn lines around them. But Dürer would likely have come to know the six designs through a series of unattributed engravings after Leonardo’s drawings during his transformative trip to Venice in 1494-95.
Print, Interlace Pattern with White Medallion, before 1521; Albrecht Dürer (German, 1471–1528); Germany; woodcut on off-white laid paper; 27.4 x 21.2 cm (10 13/16 x 8 3/8 in.) Mat: 45.7 x 35.6 cm (18 x 14 in.); Museum purchase through gift of Eleanor and Sarah Hewitt; 1944-37-1-b
The six Knots are a rare example of Dürer directly copying another artist’s work. The woodcuts appear to replicate not only the interlace configuration, but also the precise measurements of the designs, which suggests that Dürer traced the engravings. Sometime after his return to Germany, it is presumed, Dürer carefully copied the designs onto blocks of wood, which were then cut by a craftsman so expert as to be able to carve out the delicate web of knots without breaking the wood between the carved lines. Black ink would have been spread across the surface of the woodblocks, catching only the astonishingly minute recesses between the maze of lines, before the blocks were pressed against the paper to produce the prints. The Knots put Dürer in the position of both student and master. On the one hand, the project offered Dürer the opportunity to study and replicate the remarkable work of Leonardo. But the decision to reproduce Leonardo’s designs in woodcut allowed Dürer to showcase his consummate dexterity in a medium that did not ordinarily lend itself to exquisite, intricately entwined lines. Beyond enabling Dürer to demonstrate virtuosity in the woodcut, the intended use of the six prints remains an enigma. His only recorded mention of the prints, in his diary, describes his sale of a set to a glassmaker in the Netherlands.[2] Though we may never fully recover their original purpose, the beauty and complexity of the Knots continue to endure.
[1] Carmen Bambach Cappel, “Leonardo, Tagliente, and Dürer: ‘La scienza del far di groppi,’” Achademia Leonardi Vinci: Journal of Leonardo Studies & Bibliography of Vinciana 4 (1991): 72–98; Eileen Costello, “Knot(s) Made by Human Hands: Copying, Invention, and Intellect in the Work of Leonardo da Vinci and Albrecht Dürer,” Athanor XXIII (2005): 25–33; Anada Coomaraswamy, “The Iconography of Dürer’s ‘Knots’ and Leonardo’s ‘Concatenation,’” Art Quarterly 7 (Spring 1944): 109–28; Arthur M. Hind “Two Unpublished Plates of the Series of Six ‘Knots’ Engraved after Designs by Leonardo da Vinci,” Burlington Magazine 12 (October 1907/March 1908): 41–42.
[2] Albrecht Dürer, quoted in Costello, “Knot(s) Made by Human Hands,” 28.
Caitlin Condell is the Assistant Curator & Acting Head of the Drawings, Prints & Graphic Design Department.
from Cooper Hewitt, Smithsonian Design Museum http://ift.tt/2mYAKKm via IFTTT
mathematicians have a very weird standard for use of the term “well-known”
This is the most correct post about mathematicians I have ever seen.
From “Proofs Without Words” by Roger Nelsen.
Juggling has advanced enormously in recent decades, since mathematicians began systematically investigating the possible patterns of non-colliding throws. As a result of this research, many new possibilities have been discovered for jugglers to attempt. In addition, the connections between juggling and the algebra of braids provide another way to analyze juggling.
[I]n the book called Katsuyo Sanpo (“essentials of the art of calculation”) by the outstanding Japanese mathematician Takakazu Seki, published also posthumously, in 1712 (and thus 1 year before Bernoulli [posthumously published the same result]!), the formula for the sums of powers and the inductive definition of the Bernoulli numbers are given. His formula and definition are completely the same as Bernoulli’s.
Bernoulli Numbers and Zeta Functions - Arakawa & Ibukiyama & Kaneko (via eka-mark)
Exploring the Relations of Geometry, Nature and The Human Being: Geometric GIFs by Erik Söderberg.
(BoingBoing)
Mathematical and Scientific Markings on Rustic Pottery by Laura C. Hewitt
Designed in rural Alaska, artist Laura C. Hewitt’s rustic, handmade pottery is influenced by the magic found within the mundane, nature and its pragmatic obscurity. Embossed with typographical mappings of the universe, mathematical formulas, and technological charts, Hewitt’s work pays homage to the dichotomy of the union between science and art.
Adorned with patterns, which include alpha numeric marks from vintage machinist punches and inlaid drawings reminiscent of maps, circuit board, astronomy and flow charts running into deeply carved organic river markings, each piece is unique.
Chronicling binary numbers, the distance from sun, the solar year and equatorial diameter of all planets, Ohm’s law, and the mapping of a circuit board among other technical formulas, each creation is Wheel thrown and hand carved. The vintage manual typewriter keys markings add a rustic and agrarian sensibility, which create a profound juxtaposition with the numerical values of technology in Hewitt’s pottery. You can find her entire collection in her Etsy shop.
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Report Cards for Famous Mathematicians by Math With Bad Drawings
I don’t think I’d usually post these, but something about them is cute in the right kind of way. Also, I think these really succinctly speak to certain controversies / flaws in the current math education system.
Pick an x, any x, as mathemagicians would say.
Harking back to my earlier post about the SU(2) -> SO(3) double cover, I will describe one way to convince yourself that folding the n-sphere in half by identifying antipodal points leaves you with real projective n-space. Today we’ll see that this should be true just by comparing suitably-chosen definitions of both objects, and then in the next post we’ll try to actually visualize the folding-in-half process. I will try to be as non-technical as I reasonably can; comments, questions, and criticisms are encouraged.
The two basic ingredients we’ll need for our definitions are 1) the space R^n of n-tuples of real numbers (R^0 is a point, R^1 is a line, R^2 is a plane, etc.) and 2) the concept of taking a quotient by an equivalence relation (I will assume you know how this works). In both cases, we’ll actually want to remove the origin from R^n before we do anything else; this is sometimes called “puncturing” the space. We will compare a somewhat unusual definition of the n-sphere (which is equivalent to the usual definition) to the standard definition of real projective n-space.
If we remove the origin from R^(n+1), we find that each remaining point lies on exactly one ray that starts at the origin - in other words, the rays from the origin partition the rest of R^(n+1) - and we can define an equivalence relation on this punctured space which sees each ray as an equivalence class. The quotient of punctured R^(n+1) by this equivalence relation can be visualized as pushing the points within the unit n-sphere outward, and pulling the points outside the unit n-sphere inward, so that you’re left with exactly one point for each direction away from the origin in R^(n+1). This space of rays from the origin in R^(n+1) is precisely the n-sphere.
Real projective n-space is also defined as a quotient of punctured R^(n+1) . Now instead of treating each ray from the origin in the punctured space as an equivalence class, we will do the same for lines through the origin - after all, lines through the origin also partition punctured R^(n+1), and each line through the origin is just the union of two rays through the origin which point in opposite directions. This space of lines through the origin in R^(n+1) is precisely real projective n-space.
These definitions should make it clear that there are precisely two antipodal points of the n-sphere - equivalently two opposite-pointing rays from the origin in R^(n+1) - corresponding to each point of real projective n-space - equivalently each line through the origin in R^(n+1) - and that passing from the former to the latter is precisely a matter of identifying or gluing together these antipodes.