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Curved Space


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Left: The Marriage of Giovanni Arnolfini and Giovanna Cenami by Jan Van Eyck (1434), National Gallery, London
Above: Detail of Mirror

Non-Euclidean geometries have always been all around us, and were well appreciated by artists before they were recognised by mathematicians. The Arnolfini Wedding portrait shows the Tuscan merchant Giovanni Arnolfini and his wife, together with their faithful dog in their home; the entire scene is perfectly reflected in a convex mirror hanging on the wall behind them and the perspective is complicated by the use of more than one vanishing point. The fact that the logically self-consistent system of Euclidean geometry described plane shapes that could then be viewed in a distorting mirror should have suggested that the distorted view was equally consistent as a geometry on a curved surface. It is intriguing that the technique of anamorphosis used by artists from the sixteenth century onwards was also based on such distortions, but the emphasis was entirely upon the fact that the plane 'Euclidean' image can be restored by viewing at an angle or in a suitably curved mirror, rather than upon the logical consistency of the 'non-Euclidean' image.

Text from ‘The Artful Universe’ by John Barrow