To describe the very different shapes and various geometries, we are going to talk about the distance between the dots of this space. Let’s say that we went from home one kilometre to the east and one to the north. In Euclidean space the distance between start and end points of our journey equals approximately 1.414 km. We could write this as a formula and say that the square of the distance in Euclidean space is equal to the sum of the squares of the lengths of the individual segments. In geometry this statement transitions to a Pythagorean theorem and applies to triangles, but mind the fact that we are talking only just about distance between two points in space. To understand how this space looks like, imagine a imagine a sphere with a diameter of 1.273 km. Moving from a point on the equator of the sphere along the equator, you can walk one kilometre, turn left and walk another kilometre, ending up at the north pole of the sphere. The starting point of your route and the end point are now connected not by 1,414 km, but by only one kilometre of distance on the surface of the sphere. We can imagine such a path as a closed route: a kilometre to the east, a left turn, a kilometre to the north, a left turn, and after one another kilometre of travel we return to the starting point. Hyperbolic space is another special case of space in which the Pythagorean theorem does not hold: there the distance between two points is greater than the distance in flat space.

A more vivid and lifelike example of unusual distances can be seen by observing the growth of cells and plants. For example, in ordinary Euclidean space, the distance from the center of a circle to its farthest point, the radius, and the circumference are related by a simple relationship, which is known from school: the circumference is approximately 6.28 times greater than the radius. Let’s now imagine that living cells are laid out along a circle, which begin to grow and multiply, but cannot go beyond the circle: for example, we will lay out the cells along the wall of a flat round cup. Over time, loops will appear in the chain of cells, directed towards the center of the cup. The radius of our figure, consisting of living cells, is still limited by the dimensions of our cup, but the length of the chain of cells can now be up to 7-10 times greater than the radius. The patterns and spirals we see in the arrangement of sunflower seeds and cone scales are also due to this growth. Each sunflower seed grows independently, unaware of the space occupied by its neighbors, but because the overall size of the inflorescence is limited, the seeds in the center begin to shrink to make room for the other growing seeds. If we count each seed on a sunflower inflorescence as one “step,” we can find on a flat inflorescence the same triangle that we have already seen in elliptical geometry: a closed path consisting of three steps in one direction, a left turn, three steps in a perpendicular direction, a left turn, and three steps in a perpendicular direction.

Another example of a bizarre distance is the phylogenetic trees. These are diagrams, which depict different species of living beings, with lines showing the process of their evolution and origin from common ancestors. Phylogenetic trees are usually constructed by studying the genomes of related species and calculating how different they are from each other. Knowing how often changes in genes usually occur, biologists can calculate how many years ago related species were the united with the same genome. The triangle inequality applies for distances between three points on a phylogenetic tree, but an even stricter version of the inequality holds: now the distance between a pair of points must not just be less than the sum of the distances, but less than any of the distances to the third point. Such a space is called ultrametric. One of its unusual properties is that the distance from the root of the phylogenetic tree to each of its vertices is the same, despite the fact that the distance between these two vertices can be small or large. This arrangement of phylogenetic trees in evolutionary biology is called a molecular clock: the distance between two related species, which can be determined from differences in DNA molecules, seems to measure the time that has passed since the divergence of these species.

The flat Euclidean geometry and the space properties related to it seems familiar to us and it is easy to assume that this is the only possible option in space. Nevertheless, science often gives examples of spaces, where distance between points, which include both animal species, cells of living tissues, and plant seeds do not obey the known Pythagorean theorem.