Connectivity

Characterizing surfaces

In his 1854 habilitation dissertation, Riemann writes that there are types of geometric relations that do not include any metric relations (measurement) at all. The study of such geometric relations, founded by Riemann, is now known as topology. It covers geometric relations that aren't changed by smooth transformations such as twisting and bending. For example, although I can deform a sphere into a watermelon shape, I can't squeeze it into the shape of a bagel without tearing or joining it.

Watch this video for an introduction to topology and an application to analysis situs.

There are several ways of characterizing surfaces, but the one we have focused on the most is connectivity. How many cuts can you make on a surface before it is broken into two unconnected ones. Connected means that you can get from any point on a surface to any other: it is one piece. We've discussed earlier why a plane or sphere is simply connected, a cylinder is doubly connected, and a torus is triply connected.

Transcendentals

This becomes important for looking at higher transcendentals, where the qualitative shift between algebraic magnitudes and the transcendentals (such as sine) comes up in the higher degree of connectedness of the sine surface. The sine's periodicity makes it doubly connected. Similarly, the elliptical and higher transcendentals see qualitative jumps in their topological characteristics that accompany the fact that they involve magnitudes that cannot be made by the lower transcendentals. That is, elliptical functions have two types of periodicities, where the simple transcendentals (like sine and cosine) have only one. This new type of connectedness shows up in the actual magnitudes they create.

The circular transcendentals cannot be made with algebra, and trying demonstrates this. The sine of an angle φ is equal to:
φ - φ3/3! + φ5/5! - φ7/7! + ...
an expression which literally goes on forever. Just as the square root of two is not a fraction, and the sine of an angle cannot be made with algebra, the elliptical functions cannot be produced by the basic transcendentals.

In economics, the post-moon mission economy included capabilities which could not have been made previously: it was transcendental or incommensurable to what came before. Riemann’s geometric interpretation of the dynamic space of higher transcendentals provides a relatively simple way to comprehend increasing complexity.


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