Posts Tagged ‘knot theory’
“The world is bound in secret knots”*…
Everyone knows what a knot is. But knots have special significance in math and science because their properties can help unlock secrets hidden within topics ranging as widely as the biochemistry of DNA, the synthesis of new materials, and the geometry of three-dimensional spaces. In his podcast, The Joy of Wh(Y), the sensational Steven Strogatz explores the mysteries of knots with his fellow mathematicians Colin Adams and Lisa Piccirillo…
How do mathematicians distinguish different types of knots? How many different kinds of knots are there? And why do mathematicians and scientists care about knots anyway? Turns out, there’s lots of real-world applications for this branch of math, now called knot theory. It started out with the mystery of the chemical elements about 150 years ago, which were, at the time, thought to be different kinds of knots tied in the ether. Nowadays, knot theory is helping us understand how enzymes can disentangle strands of linked DNA. And also, knot theory has potential in basic research to create new kinds of medicines, including some chemotherapy drugs. But in math itself, knot theory is helping mathematicians work out the riddles of higher-dimensional spaces…
The study of knots unites the interests of researchers in fields from molecular biology to theoretical physics: “Untangling Why Knots Are Important,” from @stevenstrogatz in @QuantaMagazine. Listen here; read the transcript here.
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As we take stock of tangles, we might might send nicely-tied birthday greetings to a beneficiary and user of knot theory, Francis Collins; he was born on this date in 1950. A physician and geneticist, he discovered the genes associated with a number of diseases, led the Human Genome Project, and served as the director of the National Institutes of Health.
“A child[’s]…first geometrical discoveries are topological…If you ask him to copy a square or a triangle, he draws a closed circle”*…
Topology is the Silly Putty of mathematics. Indeed, sometimes, topology is called “rubber-sheet geometry” because topologists study the properties of shapes that don’t change when an object is stretched or distorted. As Cliff Pickover explains, this leads to the creation of some pretty confounding shapes…
Mathematicians continue to invent strange objects to test their intuitions. Alexander’s horned sphere [above] is an example of a convoluted, intertwined surface for which it is difficult to define an inside and outside. Introduced by mathematician James Waddell Alexander (1888 – 1971), Alexander’s horned sphere is formed by successively growing pairs of horns that are almost interlocked and whose end points approach each other. The initial steps of the construction can be visualized with your fingers. Move the thumb and forefinger of each of your hands close to one another, then grow a smaller thumb and forefinger on each of these, and continue this budding without limit!
Although this may be hard to visualize, Alexander’s horned sphere is homeomorphic to a ball. In this case, this means that it can be stretched into a ball without puncturing or breaking it. Perhaps it is easier to visualize the reverse: stretching the ball into the horned sphere without ripping it. The boundary is, therefore, homeomorphic to a sphere…
Read more at “The Official Alexander Sphere Appreciation Page.”
* Jean Piaget
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As we twist and turn, we might send artfully-folded birthday greetings to Sir Erik Christopher Zeeman; he was born on this date in 1925. While he is probably most-widely known as a popularizer of Catastrophe Theory, his primary contributions to math have been in topology, more particularly in geometric topology (e.g., in knot theory) and in dynamical systems. The Christopher Zeeman Medal for Communication of Mathematics of the London Mathematical Society and the Institute of Mathematics and its Applications is named in his honor.
We might also spare a thought for Satyendra Nath Bose; he died on his date in 1974. A physicist and mathematician, he collaborated with Albert Einstein to develop a theory of statistical quantum mechanics, now called Bose-Einstein statistics. Paul Dirac named the class of particles that obey Bose–Einstein statistics, bosons, after Bose.
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