#E1523D

Didactic plaques found in the National Tile Museum in Lisbon, Portugal. Top image is a geometrical schema of a pentagon followed by a geometrical schema of a pyramid. Photo by Eve Torrence.

Double helix shadow on an old dive tank in Kaka’ako, Honolulu, Hawai’i by Mark Ewald.

This statuary is located at the entrance to the German Historical Museum in Berlin. The tablet contains the “Bride’s Chair diagram“ for proving the “Pythagorean Theorem” and the diagram used for constructing the circumcenter of a triangle. Photo by Charles F. Marion.

Graph visualization is a way to discover and visualize structures in complex relations. What sort of structures are people who do large scale computation studying? We can get a glimpse by visualizing the thousands of sparse matrices submitted to the University of Florida Sparse Matrix collection using sfdp algorithm . The resulting gallery contains the drawing of graphs as represented by 2547 sparse matrices in this collection. Each of these sparse matrices (a rectangular matrix is treated as a bipartite graph) is viewed as the adjacency matrix of an undirected graph, and is laid out by a multilevel graph drawing algorithm. If the graph is disconnected, then the largest connected component is drawn. The largest graphs have tens of millions of nodes and over a billion of edges. A simple coloring scheme is used: longer edges are colored with colder colors, and short ones warmer. The graphs are in alphabetical order. Use the “Search” link to find graphs of specific characters.
via: University of Florida Sparse Matrix collection

The Nothing That Is, A Natural History Of Zero Robert Kaplan

A symbol for what is not there, an emptiness that increases any number it’s added to, an inexhaustible and indispensable paradox. As we enter the year 2000, zero is once again making its presence felt. Nothing itself, it makes possible a myriad of calculations. Indeed, without zero mathematics as we know it would not exist. And without mathematics our understanding of the universe would be vastly impoverished. But where did this nothing, this hollow circle, come from? Who created it? And what, exactly, does it mean?
Robert Kaplan’s The Nothing That Is: A Natural History of Zero begins as a mystery story, taking us back to Sumerian times, and then to Greece and India, piecing together the way the idea of a symbol for nothing evolved. Kaplan shows us just how handicapped our ancestors were in trying to figure large sums without the aid of the zero. (Try multiplying CLXIV by XXIV). Remarkably, even the Greeks, mathematically brilliant as they were, didn’t have a zero—or did they? We follow the trail to the East where, a millennium or two ago, Indian mathematicians took another crucial step. By treating zero for the first time like any other number, instead of a unique symbol, they allowed huge new leaps forward in computation, and also in our understanding of how mathematics itself works.
In the Middle Ages, this mathematical knowledge swept across western Europe via Arab traders. At first it was called “dangerous Saracen magic” and considered the Devil’s work, but it wasn’t long before merchants and bankers saw how handy this magic was, and used it to develop tools like double-entry bookkeeping. Zero quickly became an essential part of increasingly sophisticated equations, and with the invention of calculus, one could say it was a linchpin of the scientific revolution. And now even deeper layers of this thing that is nothing are coming to light: our computers speak only in zeros and ones, and modern mathematics shows that zero alone can be made to generate everything.
Robert Kaplan serves up all this history with immense zest and humor; his writing is full of anecdotes and asides, and quotations from Shakespeare to Wallace Stevens extend the book’s context far beyond the scope of scientific specialists. For Kaplan, the history of zero is a lens for looking not only into the evolution of mathematics but into very nature of human thought. He points out how the history of mathematics is a process of recursive abstraction: how once a symbol is created to represent an idea, that symbol itself gives rise to new operations that in turn lead to new ideas. The beauty of mathematics is that even though we invent it, we seem to be discovering something that already exists.
The joy of that discovery shines from Kaplan’s pages, as he ranges from Archimedes to Einstein, making fascinating connections between mathematical insights from every age and culture. A tour de force of science history, The Nothing That Is takes us through the hollow circle that leads to infinity.

via: google

Rara arithmetica; a catalogue of the arithmetics written before the year MDCI, with description of those in the library of George Arthur Plimpton, of New York (1908)
David Eugene Smith; George Arthur Plimpton

Plimpton’s mathematical library, preserved at Columbia University, may be the first specialized private collection of antiquarian scientific books formed by an American for which we have an annotated bibliographical catalogue. Published in 1908, Rara arithmetica is still widely consulted for its descriptions of fifteenth- and sixteenth-century books. Smith also discussed some of Plimpton’s early manuscripts in his History of Mathematics (Boston: Ginn & Co., 1923–25), and issued an addendum to his catalogue of Plimpton’s library in 1939

via: Archive

Anleitung zur Feldmesskunst im Nürnberger Gebiet
(Instructions for surveying, in the Nuremberg area)
Jörg Unger

via: bvbm1

Geometria theoretica et practica
Johann Baptist Roppelt, 1772

via: bvbm1 | permalink: nbn:de:bvb:22

— Thank you for sharing, nends.

Source: nends

‘Fractal’ is a word invented by Mandelbrot to bring together under one heading a large class of objects that have [played] … an historical role … in the development of pure mathematics. A great revolution of ideas separates the classical mathematics of the 19th century from the modern mathematics of the 20th. Classical mathematics had its roots in the regular geometric structures of Euclid and the continuously evolving dynamics of Newton. Modern mathematics began with Cantor’s set theory and Peano’s space-filling curve. Historically, the revolution was forced by the discovery of mathematical structures that did not fit the patterns of Euclid and Newton. These new structures were regarded … as ‘pathological,’ … as a ‘gallery of monsters,’ akin to the cubist paintings and atonal music that were upsetting established standards of taste in the arts at about the same time. The mathematicians who created the monsters regarded them as important in showing that the world of pure mathematics contains a richness of possibilities going far beyond the simple structures that they saw in Nature. Twentieth-century mathematics flowered in the belief that it had transcended completely the limitations imposed by its natural origins.
Now, as Mandelbrot points out, … Nature has played a joke on the mathematicians. The 19th-century mathematicians may not have been lacking in imagination, but Nature was not. The same pathological structures that the mathematicians invented to break loose from 19th-century naturalism turn out to be inherent in familiar objects all around us.

Freeman Dyson. Characterizing Irregularity’, Science (12 May 1978),

A4-sized poster filled with the first 10,000 decimals of phi.

Quipus or khipus (sometimes called talking knots) were recording devices used in the Inca Empire and its predecessor societies in the Andean region. A quipu usually consisted of colored, spun, and plied thread or strings from llama or alpaca hair. It could also be made of cotton cords. The cords contained numeric and other values encoded by knots in a base ten positional system. Quipus might have just a few or up to 2,000 cords.

Most of the information recorded on the quipus consists of numbers in a decimal system. In the early years of the Spanish conquest of Peru, Spanish officials often relied on the quipu to settle disputes over local tribute payments or goods production. Spanish chroniclers also concluded that quipus were used primarily as mnemonic devices to communicate and record numerical information. Quipucamayocs could be summoned to court, where their bookkeeping was recognised as valid documentation of past payments.

Some of the knots, as well as other features such as color, are thought to represent non-numeric information, which has not been deciphered. It is generally thought that the system did not include phonetic symbols analogous to letters of the alphabet. However Gary Urton (author of book ‘Signs of the Inka Khipu Binary Coding in the Andean Knotted-String Records’) has suggested that the quipus used a binary system which could record phonological or logographic data.

To date, no link has yet been found between a quipu and Quechua, the native language of the Peruvian Andes. This suggests that quipos are not a glottographic writing system and have no phonetic referent. Frank Salomon at the University of Wisconsin has argued that quipus are actually a semasiographic language, a system of representative symbols – such as music notation or numerals that relay information but isn’t directly related to the speech sounds of a particular language. The Khipu Database Project (KDP), began by Gary Urton, may have already decoded the first word from a Quipu—the name of a village, Puruchuco, which Urton believes was represented by a three-number sequence similar to a ZIP code. If this conjecture is correct, quipus is the only known example of a complex language recorded in a 3-D system.

More images via: British Museum

Model of a parabolic ring cyclide

via: Universitaets Sammlungen

Les Anaglyphes Geometriques by Henry Vuibert

found: here

Logical Problems, vol. VIII, Editura Academiei, 1981

found: here