Morwen Thistlethwaite
Encyclopedia
Morwen B. Thistlethwaite is a knot
theorist and professor of mathematics for the University of Tennessee
in Knoxville. He has made important contributions to both knot theory
, and Rubik's cube group
theory.
from the University of Cambridge
in 1967, his MSc from the University of London
in 1968 and his PhD
from the University of Manchester
in 1972 where his advisor was Michael Barratt. He studied piano
with Tanya Polunin, James Gibb and Balint Vazsonyi
, giving concerts in London before deciding to pursue a career in mathematics in 1975. He taught at the North London Polytechnic from 1975 to 1978 and the Polytechnic of the South Bank, London
from 1978 to 1987. He served as a visiting professor at the University of California, Santa Barbara
for a year before going to the University of Tennessee
, where he currently is a professor.
, which are:
Morwen Thistlethwaite, along with Louis Kauffman
and K. Murasugi proved the first two Tait conjectures
in 1987 and Thistlethwaite and William Menasco
proved the Tait flyping conjecture in 1991.
. The way the algorithm works is by restricting the positions of the cubes into groups. Each group represents a group
of cube positions that can be solved using a certain set of moves. The groups are:
The cube is solved by moving from group to group, using only moves in the current group, for example, a scrambled cube likely lies in group G0. A look up table of possible permutations is used that uses quarter turns of all faces to get the cube into group G1. Once in group G1, quarter turns of the up and down faces are disallowed in the sequences of the look-up tables, and the tables are used to get to group G2, and so on, until the cube is solved.
, developed Dowker notation
, a knot
notation suitable for computer use and derived from notations of Tait
and Gauss
.
Knot (mathematics)
In mathematics, a knot is an embedding of a circle in 3-dimensional Euclidean space, R3, considered up to continuous deformations . A crucial difference between the standard mathematical and conventional notions of a knot is that mathematical knots are closed—there are no ends to tie or untie on a...
theorist and professor of mathematics for the University of Tennessee
University of Tennessee
The University of Tennessee is a public land-grant university headquartered at Knoxville, Tennessee, United States...
in Knoxville. He has made important contributions to both knot theory
Knot theory
In topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined together so that it cannot be undone. In precise mathematical language, a knot is an embedding of a...
, and Rubik's cube group
Rubik's Cube group
The Rubik's Cube group is a mathematical group which corresponds to the set of all cube operations on Rubik's Cube, with function composition as the group operation....
theory.
Biography
Morwen Thistlethwaite received his BABachelor of Arts
A Bachelor of Arts , from the Latin artium baccalaureus, is a bachelor's degree awarded for an undergraduate course or program in either the liberal arts, the sciences, or both...
from the University of Cambridge
University of Cambridge
The University of Cambridge is a public research university located in Cambridge, United Kingdom. It is the second-oldest university in both the United Kingdom and the English-speaking world , and the seventh-oldest globally...
in 1967, his MSc from the University of London
University of London
-20th century:Shortly after 6 Burlington Gardens was vacated, the University went through a period of rapid expansion. Bedford College, Royal Holloway and the London School of Economics all joined in 1900, Regent's Park College, which had affiliated in 1841 became an official divinity school of the...
in 1968 and his PhD
Ph.D.
A Ph.D. is a Doctor of Philosophy, an academic degree.Ph.D. may also refer to:* Ph.D. , a 1980s British group*Piled Higher and Deeper, a web comic strip*PhD: Phantasy Degree, a Korean comic series* PhD Docbook renderer, an XML renderer...
from the University of Manchester
University of Manchester
The University of Manchester is a public research university located in Manchester, United Kingdom. It is a "red brick" university and a member of the Russell Group of research-intensive British universities and the N8 Group...
in 1972 where his advisor was Michael Barratt. He studied piano
Piano
The piano is a musical instrument played by means of a keyboard. It is one of the most popular instruments in the world. Widely used in classical and jazz music for solo performances, ensemble use, chamber music and accompaniment, the piano is also very popular as an aid to composing and rehearsal...
with Tanya Polunin, James Gibb and Balint Vazsonyi
Balint Vazsonyi
Balint Vazsonyi was a Hungarian pianist, international recitalist, soloist with leading orchestras, and political journalist. He made performance history in playing chronological cycles of all 32 piano sonatas by Beethoven over two days in New York, Boston, and London...
, giving concerts in London before deciding to pursue a career in mathematics in 1975. He taught at the North London Polytechnic from 1975 to 1978 and the Polytechnic of the South Bank, London
London South Bank University
London South Bank University is a university in south London. With over 25,000 students and 1,700 staff, it is based in the London Borough of Southwark, near the South Bank of the River Thames, from which it takes its name...
from 1978 to 1987. He served as a visiting professor at the University of California, Santa Barbara
University of California, Santa Barbara
The University of California, Santa Barbara, commonly known as UCSB or UC Santa Barbara, is a public research university and one of the 10 general campuses of the University of California system. The main campus is located on a site in Goleta, California, from Santa Barbara and northwest of Los...
for a year before going to the University of Tennessee
University of Tennessee
The University of Tennessee is a public land-grant university headquartered at Knoxville, Tennessee, United States...
, where he currently is a professor.
Tait conjectures
Morwen Thistlethwaite helped prove the Tait conjecturesTait conjectures
The Tait conjectures are conjectures made by Peter Guthrie Tait in his study of knots. The Tait conjectures involve concepts in knot theory such as alternating knots, chirality, and writhe...
, which are:
- Reduced alternating diagramsAlternating knotIn knot theory, a link diagram is alternating if the crossings alternate under, over, under, over, as you travel along each component of the link. A link is alternating if it has an alternating diagram....
have minimal link crossing numberCrossing number (knot theory)In the mathematical area of knot theory, the crossing number of a knot is the minimal number of crossings of any diagram of the knot. It is a knot invariant....
. - Any two reduced alternating diagramsAlternating knotIn knot theory, a link diagram is alternating if the crossings alternate under, over, under, over, as you travel along each component of the link. A link is alternating if it has an alternating diagram....
of a given knotKnot (mathematics)In mathematics, a knot is an embedding of a circle in 3-dimensional Euclidean space, R3, considered up to continuous deformations . A crucial difference between the standard mathematical and conventional notions of a knot is that mathematical knots are closed—there are no ends to tie or untie on a...
have equal writheWritheIn knot theory, the writhe is a property of an oriented link diagram. The writhe is the total number of positive crossings minus the total number of negative crossings....
. - Given any two reduced alternating diagramsAlternating knotIn knot theory, a link diagram is alternating if the crossings alternate under, over, under, over, as you travel along each component of the link. A link is alternating if it has an alternating diagram....
D1,D2 of an oriented, prime alternating link, D1 may be transformed to D2 by means of a sequence of certain simple moves called flypeFlypeIn the mathematical theory of knots, a flype is a kind of manipulation of knot and link diagramsused in the Tait flyping conjecture.It consists of twisting a part of a knot, a tangle: T by 180 degrees. Flype comes from an old Scottish word meaning to fold or to turn back. Two reduced alternating...
s. Also known as the Tait flyping conjecture.(adapted from MathWorld—A Wolfram Web Resource. http://mathworld.wolfram.com/TaitsKnotConjectures.html)
Morwen Thistlethwaite, along with Louis Kauffman
Louis Kauffman
Louis H. Kauffman is an American mathematician, topologist, and professor of Mathematics in the Department of Mathematics, Statistics, and Computer science at the University of Illinois at Chicago...
and K. Murasugi proved the first two Tait conjectures
Tait conjectures
The Tait conjectures are conjectures made by Peter Guthrie Tait in his study of knots. The Tait conjectures involve concepts in knot theory such as alternating knots, chirality, and writhe...
in 1987 and Thistlethwaite and William Menasco
William Menasco
William W. Menasco is a topologist and a professor at the University at Buffalo. He is best known for his work in knot theory.-Biography:Menasco received is B.A. from the University of California, Los Angeles in 1975, and his Ph.D. from the University of California, Berkeley in 1981, where his...
proved the Tait flyping conjecture in 1991.
Thistlethwaite's algorithm
Thistlethwaite also came up with a well-known solution to the Rubik's cubeRubik's Cube
Rubik's Cube is a 3-D mechanical puzzle invented in 1974 by Hungarian sculptor and professor of architecture Ernő Rubik.Originally called the "Magic Cube", the puzzle was licensed by Rubik to be sold by Ideal Toy Corp. in 1980 and won the German Game of the Year special award for Best Puzzle that...
. The way the algorithm works is by restricting the positions of the cubes into groups. Each group represents a group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
of cube positions that can be solved using a certain set of moves. The groups are:
- G0 = <L,R,F,B,U,D>
- This group contains all possible positions of the Rubik's cubeRubik's CubeRubik's Cube is a 3-D mechanical puzzle invented in 1974 by Hungarian sculptor and professor of architecture Ernő Rubik.Originally called the "Magic Cube", the puzzle was licensed by Rubik to be sold by Ideal Toy Corp. in 1980 and won the German Game of the Year special award for Best Puzzle that...
.- G1 = <L,R,F,B,U2,D2>
- This group contains all positions that can be reached (from the solved state) with quarter turns of the left, right, front and back sides of the Rubik's cube, but only double turns of the up and down sides.
- G2 = <L,R,F2,B2,U2,D2>
- In this group, the positions are restricted to ones that can be reached with only double turns of the front, back, up and down faces and quarter turns of the left and right faces.
- G3 = <L2,R2,F2,B2,U2,D2>
- Positions in this group can be solved using only double turns on all sides.
- G4 = {I}
- The final group contains only one position, the solved state of the cube.
The cube is solved by moving from group to group, using only moves in the current group, for example, a scrambled cube likely lies in group G0. A look up table of possible permutations is used that uses quarter turns of all faces to get the cube into group G1. Once in group G1, quarter turns of the up and down faces are disallowed in the sequences of the look-up tables, and the tables are used to get to group G2, and so on, until the cube is solved.
Dowker notation
Thistlethwaite, along with Clifford Hugh DowkerClifford Hugh Dowker
Clifford Hugh Dowker was a topologist known for his work in point-set topology and also for his contributions in category theory, sheaf theory and knot theory.-Biography:...
, developed Dowker notation
Dowker notation
thumb|200px|A knot diagram with crossings labelled for a Dowker sequenceIn the mathematical field of knot theory, the Dowker notation, also called the Dowker–Thistlethwaite notation or code, for a knot is a sequence of even integers. The notation is named after Clifford Hugh Dowker and...
, a knot
Knot (mathematics)
In mathematics, a knot is an embedding of a circle in 3-dimensional Euclidean space, R3, considered up to continuous deformations . A crucial difference between the standard mathematical and conventional notions of a knot is that mathematical knots are closed—there are no ends to tie or untie on a...
notation suitable for computer use and derived from notations of Tait
Peter Guthrie Tait
Peter Guthrie Tait FRSE was a Scottish mathematical physicist, best known for the seminal energy physics textbook Treatise on Natural Philosophy, which he co-wrote with Kelvin, and his early investigations into knot theory, which contributed to the eventual formation of topology as a mathematical...
and Gauss
Gauss
Gauss may refer to:*Carl Friedrich Gauss, German mathematician and physicist*Gauss , a unit of magnetic flux density or magnetic induction*GAUSS , a software package*Gauss , a crater on the moon...
.
External links
- http://www.math.utk.edu/~morwen/ - Morwen Thistlethwaite's home page.