But what is a cubic Hamiltonian graph, you ask? A graph, of course, is just a bunch of points (vertices) connected by lines (edges). A cubic graph is a graph where every vertex has 3 edges – that is, each vertex is connected to exactly three others in the graph. And a Hamiltonian graph is a graph which has a closed loop of edges (a cycle) that visits each vertex in the graph once and only once, (this is called a Hamiltonian cycle). So a cubic Hamiltonian graph is a graph where each vertex is joined to exactly three others and the graph has a cycle visiting each vertex exactly once.
What has made us so excited about cubic Hamiltonian graphs is watching Manning's cubic Hamiltonian graph builder in action. The builder starts from the Hamiltonian cycle in the graph. This loop of edges accounts for two out of the three edges for every vertex in the graph. The builder then starts adding in the third edge for each vertex, knitting the graph together before your eyes. The creation of a torus is particularly beautiful to watch.
Manning uses something called the LCF notation to build the graphs. This ingenious notation succinctly describes the structure of cubic Hamiltonian graphs by describing how you add the extra edge to each vertex by counting backwards or forwards around the Hamiltonian cycle. For example our torus has the LCF notation 150. This means that every vertex is joined to another 10 edges along the cycle, and this process is repeated 150 times, once for each vertex, to complete the graph. (The 1st vertex is joined to the 11th, the 2nd to the 12th, the 3rd to the 13th, and so on...)
A graph represented by the LCF notation [10,7,4,-4,-7,10,-4,7,-7,4]2 starts with a cycle of 20 vertices – inside the square brackets is a list of instructions for 10 edges, and these are repeated twice, giving the total number of extra edges, and therefore vertices, as 20. In this graph the 1st vertex is joined to the 11th (a distance of 10 edges), the 2nd to the 9th (a distance of 7), the 3rd to the 7th (a distance of 4), the 4th to the 20th (a distance of -4, or counting backwards 4 edges), the 5th to the 18th (distance of -7), and so on. This graph is knitted into a dodecahedron.
Watching Manning's program knit together these graphs is beautiful to watch. But this mathematics has many important uses as well. Not only are Hamiltonian cycles important mathematically, they also have many useful applications. You can read more about graphs on Plus, and about the role Hamiltonian cycles play in bell ringing and DNA analysis. And you can read more about Christopher Manning's work on his blog.
Today sees the launch of The Aperiodical, a new maths
magazine/blog aimed at people interested in mathematics who want to
read stuff. Aperiodical will post news stories related to maths, opinion pieces,
videos, feature articles, as well as blog posts. It will also publish accounts of monthly MathsJams and host the Carnival of Mathematics, a monthly blogging carnival.
"We started the site as a shared blogging outlet, and it grew out of
our desire to have a place on the web where we could keep up to date
with what's going on elsewhere, and to share the mathematical things
we do," says Katie Steckles, one of the editors. "We're not funded to write here, and all of the work we do on
the site is in our (increasingly rare) spare time."
"We're very keen to publish reports, exposition, videos, or anything
mathematical and interesting that you want to share. If you've got
something you want to share, or just have an idea for something,
please send it in."
Today is also the birthday of the famous mathematician Felix Klein, so
Aperiodical is running a new feature article, Klein: Outside the Bottle, as well as
a video about the Klein Bottle by stand-up mathematician Matt Parker
and editor Katie Steckles. Aperiodical is also hosting an online Google+ Hangout
today (25/4) from 6pm-6.30pm, for anyone who
wants to speak to us and find out more.
The editors of The Aperiodical are:
Peter Rowlett — Mathematician Errant, podcaster, and usually the most
bearded member of the team;
Katie Steckles — Maths Communicator, hair dye fan, and currently the
most qualified member of the team;
Christian Perfect — Group theorist, computery type, koala fan, and the
tallest member of the team.
The Plus office has opened in Barcelona! The weather is fine, the architecture is spectacular and everyone has been very friendly. And, importantly, the food is delicious! From the welcoming dinner with the conference organisers (and a delicious glass of port), to the focaccia de xocolata from the cafe round the corner to the pigs skin tapas we tried last night!
Detail of the ceiling of the nave of the Sagrada Familia in Barcelona (photo by SBA73)
I'm here as part of the Imaginary/BCN conference, inspired by the success of the Imaginary exhibition, which has, so far, been shown in twelve cities across Spain. Each city made the exhibition their own and the conference started with Ferran Dach and Maria Alberich explaining Barcelona's approach by showing the connections between maths and the art and architecture of the city. They juxtaposed photos of the vaulted ceilings of cathedrals, the corners of intersecting arches, the minarets on modern towers with images of mathematical singularities (eg those points where curves are not smooth). Maria believes that behind every feature in art and architecture is symmetry or mathematical singularity – these structures capture our attention and direct our gaze whether in a mathematical image or a work of art. They also showed a particularly striking Spanish example of maths influencing art: The swallow's tail by Salvador Dalí which was based on mathematician René Thom's catastrophe theory.
(You can read more on maths and art and architecture on Plus.)
The theme of mathematics and art continued with Maria Teresa Lozano showing a series of plaster models of mathematical surfaces made in Germany in the nineteenth century under the direction of famous mathematicians such as Felix Klein. These models are of highly complicated surfaces yet they are incredibly precise. But no one knows how they were made with such accuracy, the trade secret that seems to have vanished. It was lovely to see the forms of these surfaces appearing, consciously and unconsciously, in the work of many sculptors.
In the afternoon it was time for me to do some work and I joined Raúl Ibáñez, Andreas Loos and Thomas Vogt for a panel discussion on popularising maths. The question that stimulated the most discussion was – why are we communicating maths to the public? There were plenty of practical answers: encouraging the next generation of mathematicians, showing the maths that is part of our daily live, funding requirements. But people, including us here at Plus, have more personal reasons too. We love maths and we want to share its excitement and beauty. We want to show that maths is a creative, dynamics, human pursuit and that new mathematics is being discovered all the time. It isn't some solid stone edifice that has stood since the dawn of time - it is a beautiful astounding cathedral that is being constructed by mathematicians today and every day, yet it will never be complete. There will always be more to know, and that makes maths a very exciting world indeed.
Today we are focussing on the Imaginary exhibition itself and I'm off to a talk about the taste of mathematics, a collaboration between chefs and mathematicians!
In our Science fiction, science fact project we asked you which question from the frontiers of physics you'd most like to see answered on Plus. We have just closed the poll and with nearly 20% of your vote the winning question is Does infinity exist?. We will now go off to talk to experts on the topic, and you'll see some answers in a package of articles and podcasts to be published soon.
Meanwhile, keep voting to tell us which question you would like us to answer next!
The Science fiction, science fact project is a collaboration between Plus and FQXi, an organisation that supports and disseminates research on questions at the foundations of physics and cosmology.
The FQXi community website does for physics and cosmology what Plus does for maths:
provide the public with a deeper understanding of known and future discoveries in these areas, and their potential implications for our worldview.
The image Fractured Worlds at NASA's Kennedy Space Center, with Frank Milordi in the centre.
If, like us, you like fractals, then you will love the work of Frank Milordi, aka FAVIO. Milordi is a former Director of Engineering and Technology who creates mind challenging computer images based on the mathematics of chaos and fractals. You may be familiar with his work already, as one of his beautiful fractal images adorns one of the latest Plus postcards.
In addition to work based on those amazing infinitely repeating structures, Milordi has also created a unique form called annihilated fractal, which dominates his art. "The annihilated fractal is a transformation of the base fractal via additional mathematical manipulations," he explains. "The resulting image is a colourful and abstract form that retains elements of the base fractal. The viewer is challenged to discern elements of the base fractal within the visual abstraction."
Some favourite FAVIO images are shown below. For additional information visit his website or write to Frank Milordi, 4261 Careywood Dr, Melbourne, FL 32934 (USA).
Frank Milordi at an exhibition of his work at the Orlando Museum of Art — Shop Gallery.