(Press-News.org) Alicia Boole Stott, the third daughter of mathematician George Boole, is probably best known for establishing the term "polytope" for a convex solid in four dimensions. Alicia was also a long time collaborator of HSM Coxeter, one of the greatest geometers of the 20th Century.
Platonic solids are regular bodies in three dimensions, such as the cube and icosahedron, and have been known for millennia. They feature prominently in the natural world wherever geometry and symmetry are important, for instance in lattices and quasi-crystals, as well as fullerenes and viruses (see the recent paper by Pierre-Philippe Dechant from Durham University and Reidun Twarock's group in York [Dechant et al. (2014). Acta Cryst. A70, 162-167; doi: 10.1107/S2053273313034220].
Platonic solids have counterparts in four dimensions, and the Swiss mathematician Ludwig Schlaefli and Alicia Boole Stott showed that there are six of them, five of which have very strange symmetries. Stott had a unique intuition into the geometry of four dimensions, which she visualised via three-dimensional cross-sections.
A paper by Dechant published in Acta Crystallographica Section A: Foundations and Advances [Dechant (2013). Acta Cryst. A69, 592-602; doi: 10.1107/S0108767313021442] shows how regular convex 4-polytopes, the analogues of the Platonic solids in four dimensions, can be constructed from three-dimensional considerations concerning the Platonic solids alone. The rotations of the Platonic solids can be interpreted naturally as four-dimensional objects, known as spinors.
These in turn generate symmetry (Coxeter) groups in four dimensions, and yield the analogues of the Platonic solids in four dimensions. In particular, this spinorial construction works for any three-dimensional symmetry (Coxeter) group, such that these cases explain all "exceptional objects" in four dimensions, i.e. those four-dimensional phenomena that do not have counterparts in arbitrary higher dimensions. This also has connections to other "exceptional phenomena" via Arnold's trinities and the McKay correspondence. This spinorial understanding of four-dimensional geometry explains for the first time the strange symmetries that these four dimensional objects have.
This connection between the geometry of four dimension and that of three dimensions via rotations/spinors has not been noticed for centuries despite many of the great mathematicians working on Platonic solids and their symmetries, and is very different from Alicia Boole Stott's way of visualizing three-dimensional sections. It sheds new light on both sides, from real three-dimensional systems with polyhedral symmetries such as (quasi) crystals, viruses and fullerenes to four-dimensional geometries arising for instance in Grand Unified Theories and string and M-theory.
INFORMATION: END
Platonic solids generate their 4-dimensional analogues
2014-07-07
[Attachments] See images for this press release:
ELSE PRESS RELEASES FROM THIS DATE:
Scientists find important piece in the brain tumor puzzle
2014-07-07
This news release is available in French. Scientists at the Montreal Neurological Institute and Hospital, McGill University and McGill University Health Centre have shown that a member of the protein family known as SUMO (small ubiquitin-like modifier) is a key to why tumour cells multiply uncontrollably, especially in the case of glioblastoma. The SUMO family proteins modify other proteins and the SUMOylation of proteins are critical for many cellular processes. Identifying SUMO's role in the cancer cell growth will lead to a new strategy for glioblastoma treatment. ...
NASA sees Hurricane Arthur's July fourth fireworks on US East Coast
2014-07-07
VIDEO:
On July 3, NASA's TRMM satellite passed over Arthur and saw intense bands of thunderstorms north of Arthur's well defined eye dropping rainfall at a rate of over 98.4 mm...
Click here for more information.
Hurricane Arthur made landfall in North Carolina on July 3, and today, July 4, it is bringing its own fireworks along the Mid-Atlantic and New England states. Those fireworks were in the form of "hot towers," powerful, high thunderstorms with heavy rainfall that indicate ...
Conclusion of the Lindau Nobel Laureate meeting
2014-07-07
The 64th Lindau Nobel Laureate Meeting ended with a panel discussion entitled "Science for the benefit of mankind" on Mainau Island, Lake Constance, Germany, today." 37 Nobel laureates and more than 600 selected young scientists from 80 countries had participated in the week-long meeting in the Bavarian city of Lindau since last Sunday. A boat trip to Mainau at the invitation of the State of Baden-Württemberg marked the finish of the programme which was devoted to medicine and physiology. In 1895 Alfred Nobel determined in his will that the Nobel Prizes shall be awarded ...
New optogenetic tool for controlling neuronal signalling by blue light
2014-07-07
Institute for Basic Science (IBS), the main organization of the International Science and Business Belt project in South Korea, has announced that a group of researchers, led by professor Won Do Heo, have developed a new technology in the field of optogenetics that can remotely control specific receptors by light. They have named this new technology "OptoTrk" and it has succeeded with neuronal differentiation inducement.
The most significant feature of OptoTrk technology is that it requires only light to activate neuronal functions without the need of other substances. ...
Perspective of the PandaX dark matter experiment
2014-07-07
The PandaX experiment of China, which is located in the deepest underground laboratory, has released its technical design report recently. The full article will appear in SCIENCE CHINA Physics, Mechanics & Astronomy, 2014, 57(8): 1476-1494.
The Particle and Astrophysical Xenon (PandaX) collaboration was established in 2009 and mainly supported by the Ministry of Science and Technology, the Ministry of Education in China， the Natural Science Foundation of China，and Shanghai Jiao Tong University. The experiment is suitable for both direct dark matter detection ...
New type of soot particle discovered from wildfire emissions
2014-07-07
RENO – Every year, wildfires clear millions of hectares of land and emit around 34-percent of global soot mass into the atmosphere. In certain regions, such as Southeast Asia and Russia, these fires can contribute as much as 63-percent of regional soot mass.
In a paper published in Nature Scientific Reports, a team of scientists led by Rajan Chakrabarty from Nevada's Desert Research Institute report the observation of a previously unrecognized form of soot particle, identified by the authors as "superaggregates," from wildfire emissions. These newly identified particles ...
Blocking cells' movement to stop the spread of cancer
2014-07-07
Insights into how cells move through the body could lead to innovative techniques to stop cancer cells from spreading and causing secondary tumours, according to new UCL research.
Scientists discovered that cells can change into an invasive, liquid-like state to readily navigate the narrow channels in our body. This transformation is triggered by chemical signals, which could be blocked in order to stop cancer cells from spreading.
Most cancer deaths are not due to primary tumours, but to secondary tumours in vital organs, such as the lungs or brain, caused by cells ...
Do you look infected? Should I kill you? No, I'm fine, move along
2014-07-07
Some viruses can hide in our bodies for decades. They make 'fake' human proteins that trick our immune cells into thinking 'everything is awesome', there's nothing to see here.
Now researchers at the Imaging Centre of Excellence at Monash and Melbourne Universities have determined the basic structure of one of the two known families of these deceptive proteins.
Using synchrotron light and working with a common virus that lives in people happily and for the most part harmlessly, they worked out the structure of the fake proteins. This is an important first step towards ...
NUS researchers discover novel protein complex with potential to combat gastric cancer
2014-07-07
A team of scientists from the Cancer Science Institute of Singapore (CSI Singapore) at the National University of Singapore (NUS) discovered that a protein named IL23A is part of our stomach's defence against bacterial infection which leads to gastric cancer. This finding could potentially be used to combat the deadly disease.
The research group, led by Professor Yoshiaki Ito, Senior Principal Investigator at CSI Singapore, also showed that the production of IL23A by stomach cells requires the tumour suppressor gene, RUNX3, which is frequently silenced in gastric cancer. ...
Water bonus flows from climate change measures
2014-07-07
The equivalent of one-third of Melbourne's water use could be saved each year through the implementation of efficiency measures that deal with climate change, according to a new study.
Researchers at the Monash Sustainability Institute analysed the water-saving potential of 74 options for reducing greenhouse gas emissions identified in ClimateWorks Australia's award-winning Low Carbon Growth Plan for Australia. The research was published this week in Springer's international journal Climatic Change.
Monash University Research Fellow Dr Philip Wallis said an analysis ...