Nothing better illustrates the difference between the intuitive understanding that something is true and being convinced by logic that something is true than the visual proof of the theorem egregium of Gauss found in “Visual Differential Geometry and Forms” by Tristan Needham and the 9 step algebraic proof in “The Geometry of Spacetime” by Jim Callahan.

Mathematicians attempt to tie down the Gulliver of our powerful appreciation of space with Lilliputian strands of logic.

First: some background on the neurology of vision and our perception of space and why it is so compelling to us.

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). This wasn’t 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. Even so, it was clear that the occipital lobe was important — destroy it on both sides and you are blind — https://en.wikipedia.org/wiki/Occipital_lobe but the occipital lobe accounts for 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. Over 30 retinal cell types in our retinas are known, each responding to a different aspect of the visual stimulus. For instance, there are cells responding to color, to movement in one direction, to a light stimulus turning on, to a light stimulus turning off, etc. etc.

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) is able to show us increased neuronal activity primarily by the increase in blood flow it causes.

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

I’ve written several posts about them.

*is*the theorem egregium? Look at any object (say a banana). You can see how curved it is by just looking at its surface (e.g. how it looks in the 3 dimensional space of our existence). Gauss showed that you don’t

*can’t*get outside of). Is our universe curved or not? To study the question is to determine its intrinsic curvature by extrapolating the tools Gauss gave us to higher dimensions and comparing the mathematical results with experimental observation. The elephant in the room is general relativity which would be impossible without this (which is why I’m studying the theorem egregium in the first place).

*is*. But you have a clear definition in terms of the ratio of areas, which soon disappears in a massive (but necessary) algebraic fandango.

**although I believe it, I don’t see it**. This certainly isn’t to denigrate Callahan, who has helped me through his book, and a guy who I consider a friend as I’ve drunk beer with him and his wife while listening to Irish music in a dive bar north of Amherst.

*not*the circumference as I thought I remembered). Since the circumference of a circle is 2*pi*radius, radian measure varies from 0 to 2*pi. So a right angle is pi/2 radians.

*comprehensible geometrical expositions*, of geodesics, holonomy, parallel transport and vector fields, and you should have mental images of them all.Interested? Be prepared to work, and to reorient how you think about these things if you’ve met them before. The 3 links mentioned about will give you a glimpse of Needham’s style. You probably should read them next.