Visual Differential Geometry and Forms — Take 3

Visual Differential Geometry and Forms is a terrific math book about which I’ve written

here —


here —

but first some neurology.

In the old days we neurologists figured out what the brain was doing by studying what was lost when parts of the brain were destroyed (usually by strokes, but sometimes by tumors or trauma).  Not terribly logical, as pulling the plug on a lamp plunges you in darkness, but the plug has nothing to do with how the lightbulb or LED produces light.  It was clear that the occipital lobe was important — destroy it on both sides and you are blind — but it’s only 10% of the gray matter of the cerebral cortex.

The information flowing into your brain from your eyes is enormous.  The optic nerve connecting the eyeball to the brain has a million fibers, and they can fire ‘up to 500 times a second.  If each firing (nerve impulse) is a bit, then that’s an information flow into your brain of a gigaBit/second.   This information is highly processed by the neurons and receptors in the 10 layers of the retina.  There are many different cell types — cells responding to color, to movement in one direction, to a light stimulus turning on, to a light stimulus turning off, etc. etc.   Over 30 cell types have been described, each responding to a different aspect of the visual stimulus.

So how does the relatively small occipital lobe deal with this? It doesn’t.At least half of your the brain responds to visual stimuli.  How do we know?   It’s complicated, but something called functional Magnetic Resonance Imaging (fMRI) picks up increased neuronal activity (primarily by the increase in blood flow it causes).

Given that half your brain is processing what you see, it makes sense to use it to ‘see’ what’s going on in Mathematics.  This is where Tristan Needham’s books come in.

If you’ve studied math at the college level with some calculus you shouldn’t have much trouble.  But you definitely need to look at Euclid as it’s used heavily throughout. Much use is made of similar triangles to derive relationships.

I’ll assume you’ve read the first two posts mentioned above.  Needham’s description of curvature and torsion of curves in 3 dimensional space is terrific.  They play a huge role in relativity, and I was able to mouth the formulas for them but they remained incomprehensible to me, as they are just symbols on the page.  Hopefully the discussion further on in the book will let me ‘see’ what they are when it comes to tensors.

He does skip about a bit passing Euclid when he tell how people living t on a 2 dimensional surface could tell it wasn’t flat (p. 19).  It involves numerically measuring the circumference of a circle.  But this involves a metric and putting numbers on lines, something Euclid never did.

Things really get interesting when he started talking about how Newton found the center of curvature of an arbitrary curve.  Typically Needham doesn’t really define curve, something obvious to a geometer, but it’s clear the curve is continuous.  Later he lets it slip that the curve is differentiable (without saying so).

So what did Newton do?   Start with a point p and find its normal line.  Then find point q near p on the curve and find its normal line and see where they intersect.  The center of curvature at p is the point of intersection of the normals as the points get closer and closer to p.

This made wonder how Newton could find the normal to an arbitrary continuous curve.  It would be easy if he knew the tangents, because Euclid very early on (Proposition 11 Book 1) tells you how to construct the perpendicular to a straight line.  It is easy for Euclid to find the tangent to a circle at point p — it’s just the perpendicular to the line formed between the center of the circle (where you put one point of the compass used by Newton) and the circle itself (the other point of the compass.

But how does Newton find the tangent to an arbitrary continuous curve?  I couldn’t find any place that he did it, but calc. 101 says that you just find the limit of secants ending at p as the other point gets closer and closer.  Clearly this is a derivative of sorts.

Finally Needham tells you that his curves in 3 dimensions are differentiable in the following oblique way.  On p. 106 he says that “each infinitesimal segment (of a curve) nevertheless lies in a plane.”  This tells you that the curve has a tangent, and a normal to it at 90 degrees (but not necessarily in the plane).  So it must be differentiable (oblique no?).   On p. 107 he differentiates the tangent in the infinitesimal plane to get the principal normal (which DOES  lie in the plane).  Shades of Bishop Berkeley (form Berkeley California is named) — differentiating the ghost of a departed object.

Addendum: 28 March ’22:  It’s was impossible for me to find a definition of surface even reading the first 164 pages.  Needham highly recommends a book I own “Elementary Differential Geometry (revised 2nd edition) by Barrett O’Neill calling it “the single most clear-eyed elegant, and (ironically) modern treatment of the subject  . . .  at the undergraduate level”.  The first edition was 1966.   In the preface to his book O’Neill says “One weakness of classical differential geometry is its lack of any adequate definition of surface”.    No wonder I had trouble.

So it’s great fun going through the book, get to “Einstein’s Curved Spacetime” Chapter 30 p. 307, “The Einstein FIeld Equation (with Matter) in Geometrical Form” complete with picture p. 326.

More Later.

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