Tests given to an Amazonian tribe called the Mundurucu suggest that our intuitions about geometry are innate. Researchers examined how the Mundurucu think about lines, points and angles, comparing the results with equivalent tests on French and US schoolchildren. The Mundurucu showed comparable understanding, and even outperformed the students on tasks that asked about forms on spherical surfaces.

A high-flying balloon that soared over Antarctica has answered one of cosmology's greatest questions by revealing that the fabric of the Universe is "flat". To astronomers, flat means that the usual rules of geometry are observed - light not being bent by gravity travels in straight lines, not curves. But since Albert Einstein proposed that the Universe may be "curved", the debate has been open.

Researchers have developed a simple technique that adds evidence to the theory that the Universe is flat. Moreover, the method - developed by revisiting a 30-year-old idea - confirms that "dark energy" makes up nearly three-quarters of the Universe. The research, published in Nature, uses existing data and relies on fewer assumptions than current approaches.

Benoit Mandelbrot, who discovered mathematical shapes known as fractals, has died of cancer at the age of 85. Mandelbrot, who had joint French and US nationality, developed fractals as a mathematical way of understanding the infinite complexity of nature.

The way fungus-like slime moulds grow could help engineers design wireless communication networks. Scientists drew this conclusion after observing a slime mould as it grew into a network that was almost identical to the Tokyo rail system.

It is called the Shimura-Taniyama-Weil (STW) conjecture, and it has baffled and defeated some of the greatest minds in maths over the last 40 years. The STW conjecture links two seemingly unrelated areas of mathematics: the theory of numbers and the theory of shapes or, as mathematicians prefer to call them, elliptic curves and modular forms. For decades, mathematicians have studied these subjects realising that there are deep connections between them but without ever being able to pin down the exact relationship. Andrew Wiles used the STW conjecture to provide the proof for the famous mathematical puzzle Fermat's Last Theorem. But before Wiles cracked the theorem in 1993, nobody even knew where to begin to tackle the STW conjecture. In fact, Fermat's Last Theorem is only a particular part of the more profound STW conjecture. In technical terms, Andrew Wiles, a professor at Princeton, proved the STW conjecture for what are called semi-stable elliptic curves.

A study of medieval Islamic art has shown some of its geometric patterns use principles established centuries later by modern mathematicians. Researchers in the US have found 15th Century examples that use the concept of quasicrystalline geometry. This indicates intuitive understanding of complex mathematical formulae, even if the artisans had not worked out the underlying theory, the study says.

What's attractive about studying E8 is that it's as complicated as symmetry can get. Mathematics can almost always offer another example that's harder than the one you're looking at now, but for Lie groups, E8 is the hardest one. ... Each of the 205,263,363,600 entries on the matrix is far more complicated than a straightforward number; some are complex equations.

Excellent about the need for re-unification of mathematics and geometry, and physics. ... Also: Jacobi noted, as mathematics' most fascinating property, that in it one and the same function controls both the presentations of a whole number as a sum of four squares and the real movement of a pendulum. These discoveries of connections between heterogeneous mathematical objects can be compared with the discovery of the connection between electricity and magnetism in physics or with the discovery of the similarity between the east coast of America and the west coast of Africa in geology. The emotional significance of such discoveries for teaching is difficult to overestimate. It is they who teach us to search and find such wonderful phenomena of harmony of the Universe.

A Multi-Disciplinary Exploration of Non-Orientable Surfaces