From Impossible Shapes to the Nobel Prize

Roger Penrose, British mathematical physicist, mathematician and philosopher of science has just been named as one of the winners of the 2020 Nobel Prize in Physics. Penrose has made major contributions to general relativity and cosmology.

Impossible triangle sculpture in Perth, Western Australia [image Wikimedia Commons].

Penrose has also come up with some ingenious mathematical inventions. He discovered a way of defining a pseudo-inverse for matrices that are singular, he rediscovered an “impossible object”, now called the Penrose Triangle, and he discovered that the plane could be tiled in a non-periodic way using two simple polygonal shapes called kites and darts.

The Tribar

In 1954, while Penrose was at a conference in Amsterdam, he happened upon an exhibition of the remarkable art work of E. C. Escher. This inspired him to construct the “impossible figure” which is now called the Penrose Triangle or Penrose Tribar. It is a shape that looks like a solid three-dimensional triangular form, but that is impossible to realise in 3D Euclidean space.

The Penrose triangle or tribar [image from Wikimedia Commons].

The tribar appears to be a solid triangular object made from three straight sections, each with square cross-section, meeting at right angles to each other. It cannot be realized as a closed loop in our 3D space. It is an illusion.

However, a construction with three straight square bars can be made that, when viewed from a certain angle, appears to be a tribar. There is an “Impossible Triangle sculpture in Perth, Western Australia (see figure at the top of this post). Seen from the correct angle, it appears to be a Penrose triangle.

In fact, the shape had already been discovered in 1934 by a Swedish artist Oscar Reutersvärd, but had apparently attracted little notice at the time. Together with his father Lionel, Penrose popularised the shape, and it has since appeared in countless books on art, psychology and mathematics, and elsewhere. Penrose has described it as “impossibility in its purest form”.

Penrose sent a copy of an article on the shape to Escher, who was strongly influenced by it. Escher’s well-known masterpiece Waterfall shows water flowing over a mill-wheel and returning in a zig-zag channel to the top again. It incorporates the structure of the tribar in an essential way.

Another illusion devised by Penrose is the curious cyclic staircase shown here.  It appears to descend clockwise and ascent anticlockwise, and yet we arrive back at the same spot after one cycle, whichever way we go. This also inspired Escher to produce a lithograph, Ascending and Descending, which depicts a never-ending staircase on the roof of a large building.

The 2020 Physics Nobel Prize

Roger Penrose has been awarded half of the 2020 Nobel Prize in Physics for work first presented in his 1965 paper “Gravitational Collapse and Space-Time Singularities”.

The prize was awarded to Penrose for showing how black holes can form. He shares the prize with two astrophysicists who discovering a black hole at the centre of the Milky Way. The citation stated that Penrose showed that “black hole formation is a robust prediction of the general theory of relativity”

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