Past Escher’s edge.

Late in the 1950s the mathematician H. S. M. Coxeter sent the Dutch artist M. C. Escher a paper. In it was a pattern of triangles that grew smaller and smaller toward the edge of a circle and never reached it. It gave Escher, in his words, “quite a shock”: here was a way to fit an endless pattern inside a circle. Between 1958 and 1960 he made four prints from the idea. The last, Circle Limit IV, fills its circle with white angels and black devils.

To anyone living in that circle, every angel is the same size, and so is every devil. They only look smaller toward the edge because a whole endless world has been drawn inside a circle. Mathematicians call this way of drawing it Poincaré’s disk. Its edge, the rim, is endlessly far from the centre: walk toward it for ever and you never arrive. Small shapes are kept and sizes are not, so a devil’s wings meet at the same angle wherever it flies.

In an interview posted in 2024, Roger Penrose reached for these prints to explain the word at the heart of his picture of the universe. Conformal, he said, means that small shapes are kept while size is not: the angle of a devil’s wings is the same however near the edge you go. It is the word behind his idea that the end of one universe could become the start of the next.

Our index has a crossing of its own. Every look a mind hands back crosses into it carrying what was found, and no count of how often. So we set our index beside Escher’s world and moved that world a step at a time, to see what crosses his rim and what changes as it moves.

Our index gives every concept on its map a place on a circle. By our index’s own rule, how close two concepts read comes from those two places and nothing else: the nearer they sit, the closer they read. Escher’s rim is a circle too. So we stood three concepts on it, A, B and C, and asked what each move of his world does to how close they read.

Escher’s print stops at the rim. Its geometry does not. Mirror the world in its own rim, as if the rim were a mirror bent into a circle. Every point goes straight out from the centre along its own line, to its matching place outside. The tiling goes on past the rim into a second endless world, the same as the first. Nothing on the rim moves at all.

So every closeness is kept: each concept keeps its direction from the centre. We proved more. Picture the flat page Escher’s circle is drawn on, running on for ever. Shift it, turn it, zoom it, flip it or mirror it in a circle, alone or combined, and every angle on it is kept. Of all those moves, exactly these keep how close every two concepts read: the turns about the centre, the mirrors through it, the zooms in or out from it, and the mirror in the rim, alone or combined. The mirror in the rim turns the page inside out: what was near the centre goes far away, and what was far away comes near.

Inside Escher’s world you can also slide the whole picture along one of its own straight lines, the way a step along a corridor carries you. Every angel keeps its true size and shape. On the page, the tiles on one side shrink toward the rim, and those on the other side come in from it and grow.

This time the closeness changes. B and C run round the rim toward A, and every pair reads differently. That a slide changes something is not down to the three we chose. We proved it for any three concepts on the rim, as long as no two share a place or sit opposite. Take the mixes of shifts, turns, zooms, flips and mirrors that carry Escher’s world onto itself, its slides among them. Of those, exactly the turns about the centre and the mirrors through it keep every closeness among the three. Every slide changes at least one.

Our index reads every closeness round one circle, from its centre. A slide carries the world past that centre: what stood there moves away, and something else arrives. The disk’s own distances are all kept, and still the view from the centre changes. Piece 7 says something like it of every look in the universe: there is no view from nowhere.

One more thing can cross. Put a spring in Escher’s world, flowing out evenly on every side, like a lamp in a flat land. Draw any loop around it and the same amount flows out through the loop: all that the spring gives. Mirror the world in its rim, and the spring arrives outside with all of its flow. What crosses the rim is the whole amount.

Penrose’s handover carries an amount something like this. In a paper posted in 2025, he and Krzysztof Meissner carry mass from one universe into the next as a total flow: the flow of a field built from gravity through a closed surface around it. That total could change only if something crossed the surface.

Here our index and Escher’s world part. Our crossing keeps what was found and sheds how often it was found: a look carries no count. Escher’s rim carries the spring’s whole amount across. The two share the move, keeping the shape across the crossing, and carry different things over it.

All of this happens on a flat page, in two dimensions. Our concepts stand on the circle, where our index places them, and the proof about slides holds there, on the rim. It tells us how closeness behaves under the moves of Escher’s world. Whether our own universe cycles as Penrose proposes, it does not say: that stays his question, and a likeness we love.