Phase Space

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‘I don’t get it,’ Philmus said. ‘What other pole? How can a sphere have two centres?’

‘Think about the equator,’ whispered Himmelfarb. ‘The globe of Earth, remember? As you travel north, as you pass the equator, the concentric circles of latitude start to grow smaller, while still enclosing those to the south …’

‘We aren’t on the surface of a globe.’

‘But we are on the surface of a 3-sphere – the three-dimensional surface of a four-dimensional hypersphere. Do you see? The concentric spheres you see are exactly analogous to the lines of latitude on the two-dimensional surface of a globe. And just as, if you stand on the equator of Earth, you can look back to the south pole or forward to the north pole, so here, at the universe’s equator, we can look towards the poles of Earth or God. The Primum Mobile, the equator of the universe, curves around the Earth, below us, and at the same time it curves around God, above us.’

Philmus looked back and forth, from God to Earth, and she saw, incredibly, that Himmelfarb was right. The Primum Mobile curved two ways at once.

The Monsignor’s jaw seemed to be hanging open. ‘And Dante saw this? A four-dimensional artefact? He described it?’

‘As remarkable as it seems – yes,’ said Himmelfarb. ‘Read the poem again if you don’t believe it: around the year 1320 Dante Alighiero wrote down a precise description of the experience of travelling through a 3-sphere. When I figured this out, I couldn’t believe it myself. It was like finding a revolver in a layer of dinosaur fossils.’