History of Geometry IV: The emergence of higher dimensions | Sociology and Pure Maths| NJ Wildberger

In this history of mathematics, the 19th century stands out as an especially important chapter in the story of geometry. One of the key developments here is the move to understanding and studying higher dimensions. Here we touch on some of these advances, with an aim to explaining: where do these higher dimensions come from?

We mention projective geometry, cyclography, spaces of lines and conics and more.

Video Content:
00:00 Introduction
2:45 Motivations for higher dimensional geometry
6:08 The projective line
12:15 Projective space
16:45 Space of circles in 2D (cyclography)
19:50 Space of lines in 3D (Plucker)
27:44 Space of curves (planar)

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[rbrowne]

This fellow is addictive! I got up today and made myself a coffee (4D, 1e&1f redundant), and saw a recommendation on utube for him. Voila! I watched the whole thing! Fascinating.

[tyscott]

Great thought process, I love it. I was thinking in a higher dimension when I saw your drumstick as a pointer and wondered if you took it from a student for not paying attention.

[Achrononmaster]

Mention of Schläfli polytopes made me think of Nima Arkani-Hamed's Amplituhedra. I have to tackle that stuff. Thanks.

[Larootan]

Thank you, Norman. Love your lectures!

[rgerk]

Is there a maxel of the dimension of the curve by the dimension of the space that it is in?

[alexakalennon]

Just realized, we learnt the equation for lines in 2d und 3d in school. And various intersections between planes and lines and the sphere. Never new where it originated from.

[santerisatama]

I get that "point" means here - at least mostly - numerable algebraic point in the context of a enumarated coordinate system. But when we think of coordinate systems as kinds of graphs, then more correct term for a meet of lines(/curves/edges) of a coordinate system graph would be more consistently called "vertex" than "point". By calling "vertex" a "point", all connection to Euclid is lost, Elementa gives perfectly sensible definition of geometric 'point' in postulates 1 and 3, with intuition that a point is intuitively observable only on a flat plane as the end of a straight line when all lines are parallel, otherwise only lines etc. can be observed by a flatlander, not a point. Let's also keep in mind that a dot - an area - is not really a point.

0-dimension is an alternative way of expressing that "point has no part", but in context of Elementa and sound intuition, it doesn't make any sense to say that a line can have more than 2 points, or that line consists of - infinity! - of points. A line has no (end)points, a ray has 1 point, and a segment 2 points, and that's it. On the other hand, if we try to avoid semantic and terminological confusion, we can observe and say that vertices, meets of lines, can divide lines etc. into segments.

After coordinate systems and curvatures came into play of Post-Euclidean geometry, things got messy - which is not a bad thing as such, evolution is messy. Sadly Hilbert's geometry - where "point" is postulated as an undefined "primitive notion" - only made things worse. Whitehead must have been aware of the main cause of the mess, as after the failure of Principia Mathematica as a solid foundation, he started to tentatively talk about 'point-free geometry'.

Wildberger's approach is a big step forward - he discusses the problem of continuum in honest way, and doesn't try to claim that a line etc. can actually consist of points, on the contrary. Point-reductionism is at the root of the folly of "real numbers".

There is still messy baggage to clarify. Speaking of vertices instead of points when we proceed beyond Elementa could avoid much of unnecessary confusion, IMHO.

[avinashsparrow]

Sir do make video on how to self taught beauty of maths, if u want to know higher subjects like ...group theory, tensor, operators

[theoremus]

It is interesting to note that Hamilton discovered that rotations in three dimensions required a fourth parameter, quaternion.

[jnhrtmn]

Awesome, but I think physicists overuse math. They let it dictate too much. Dimensions 1 and 2 do not exist without 3 dimensions, so why does the 4th? Theoretically, they make a great analogy, but I think "change" finds extremes at right angles, so math is there to exploit that right angle thing. Then, I don't think change needs time as the 4th dimension. Time is a math marker that accounts for changes -accounting. You could use Oreo cookies for that, just count the cookies that represent a change. I was stuck on what constitutes a rate for a while, but I don't think that means anything either. Cause and effect are "ABSOLUTE". All observers agree. I can ruin angular momentum for you if you let me. Everyone got that from math, and it does not even exist.