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Peirce fits the whole Earth into a square. The North Pole sits in the middle, the equator forms a smaller square turned through 45°, and the South Pole breaks up into the map's four corners. The projection is conformal, so the 24 uncut Tissot indicatrices are perfect circles. Their diameter depends on position: at 60° it is only 7% larger than at the North Pole, at 30° about 30% larger, and on the equator half as large again. Circles on the equator therefore cover more than twice the area of those near the pole.
The other six ellipses lie at 90° east and west. Two of them, on the equator, land exactly on the midpoints of the square's left and right sides. These are two of the four points where conformality fails and the scale grows without limit. The remaining four, at 30°S and 60°S, are crossed by a cut running from such a point to a corner of the map, so each is split into two halves in different places.
Notice the symmetry. Circles at 60°S have the same diameter as those at 60°N, because both hemispheres are projected the same way. Poland, at 52°N, is enlarged by about a quarter compared with the pole. Drag it to 72°N or 72°S and in both places it is almost at the pole's scale, although in the south it lands in a corner. Move it to the equator near Gabon and it grows sharply, because it approaches the singular point in the middle of the bottom side. The arrows of the true-north field point roughly toward the middle of the square from every side, so which way is “up” depends on where you look.
In the exact center of the square. The equator forms a smaller square turned by 45°, and the South Pole splits into the four corners of the map. That is why the direction up depends on where you stand.
The diameter of Tissot circles at 60° is only 7% larger than at the North Pole, at 30° about 30% and on the equator by half. The area of the circles on the equator is therefore more than twice that at the pole.
The projection is conformal, so the 24 uncut Tissot ellipses are perfect circles. Angle fidelity fails only at four singular points, where the scale grows without limit.
Near the equator around Gabon a country grows sharply, because it approaches the singular point in the middle of the lower side of the square. Poland at 72°N or 72°S stays close to the polar scale, though in the south it lands in a corner.
