Category Archives: Math

Book Review: Proving Ground, Kathy Kleiman

Proving Ground is a fascinating book about the 6 women who programmed the first programmable computer, the ENIAC (Electronic Numerical Integrator And Computer).  Prior to this, the women were computers as the term was used in the 1940s for people who sat in front of calculating machines and performed lengthy numerical computations solving differential equations to find the path of an artillery shell one bloody addition/subtraction/multiplication/division at a time.  When World War II started and when the man were off in the army, the search was on for  women with a mathematical background who could do this.

A single trajectory took a day to calculate, and each trajectory had to be separately calculated for different wind currents, air temperature, speed and weight of the shell.  The computations were largely done at the Moore School of Engineering at Penn and were way too slow (although accurate) to produce the numbers of trajectories the army needed.

Enter Dr. John Mauchley who had an idea of how to do this using vacuum tubes, and a brilliant 23 year old engineer, J. Presper Eckert, who could instantiate it. The army committed money to building the machine, which came in 42 monster boxes 8 feet tall, 2 feet wide and what looks like 4 feet deep.

6 of the best and brightest computers of trajectories were recruited to figure out how to wire the boxes together to mimic the trajectory calculations they had already been doing.  So, if you’ve ever done any programming, you’ll know that having a definite target to mimic with software makes life much easier.

Going a bit deeper, if you’ve done any programming in machine language, you know about registers, the addition and logical unit, hard wired memory, alterable memory.

Here’s what the 6 women were given by Dr. Eckert (without ever seeing the monster boxes)

l. A circuit diagram of each box, showing how this vacuum tube activated that vacuum tube etc. etc. The 42 boxes contained 18,000 vacuum tubes.  Vacuum tubes and transistors are similar in that their utility is that they only conduct electricity in one direction and can be turned on and off.

2. A block diagram — which showed how the functions of a unit or system interrelate

3. A logical diagram — places for dials switches, plug and cables on the front of the 42 units.

So given this, the 6 had to figure out what each unit did, and how to wire them together to mimic the trajectory calculations they had been doing.

They did it, and initially without being able to enter the room with the boxes (because they didn’t have the proper security clearance).  Eventually they got it and were able to figure out how to wire the boxes together.

If that isn’t brilliant enough, because the calculations were still taking too long, they invented parallel programing.

For those of you who know computing, that should be enough to make you thirst for more detail.

The book contains a lot of sociology.  The women were treated like dirt by the higher ups (but not by Mauchley or Eckert).  When the time came to show ENIAC off to the brass (both academic and military), they were tasked with serving coffee and hanging up coats.  When Kleiman found pictures of them with ENIAC and asked who they were, she was told they were ‘refrigerator ladies’ — whose function was similar to the barely clothed models draped over high powered automobiles to sell them.

I’ll skip the book’s sociology for some sociology of my own.  The book has biographies and much fascinating detail about all 6 women.  I grew up near Philly, and know the buildings at Penn where this was done (I went to Penn Med). Two of the 6 were graduates of Chestnut Hill College, a small Catholic school west of Philly.  The girl across the street went there.  Her mother was born in County Donegal and cleaned houses.  Her father dropped out of high school at 16 to support his widowed mother.  No social services between the two world wars, wasn’t that terrible etc. etc.  Her father worked in a lumberyard, yet the two of them sent both children to college, and owned their own home (eventually free of debt).  The Chestnut Hill grad I know became an editor at Harcourt Brace, her brother became a millionaire insurance executive.  It would be impossible for two working class people to do this today where I grew up (or probably in most places).

What is dx really?

“Differential geometry is the study of properties that are invariant under change of notion”  — Preface p. vii of “Introduction to Smooth Manifolds” J. M. Lee second edition.  Lee says this is “funny primarily because it is so close to the truth”.

Having ascended to a substantial base camp for the assault on the Einstein Field equations (e.g. understanding the Riemann curvature tensor), I thought I’d take a break and follow Needham’s advice about another math book “Elementary Differential Geometry” 2nd edition (revised 2006) by Barrett O’Neill.  “First published in 1966, this trail-blazing text pioneered the use of Forms at the undergraduate level.  Today, more than a half-century later, O’Neill’s work remains, in my view, the single most clear-eyed, elegant and (ironically) modern treatment of the subject available — present company excepted! — at the undergraduate level.”

Anyone taking calculus has seen plenty of dx’s — in derivatives, in integrals etc. etc..  They’re rarely explained.  O’Neill will get you there in just the first 24 pages.  One more page and you’ll understand

df =  (partial f/partial x) * dx + (partial f/partial y) * dy + (partial f/partial z) * dz

which you’ve doubtless seen before primarily as (heiroglyphics) before you moved on.

Is it easy?  No, not unless you read definitions and internalize them immediately.  The definitions are very clearly explained.

His definition of vector is a bit different — two points in Euclidean 3-space (written R^3 which is the only space he talks about in the first 25 pages).  His 3-space is actually a vector space in which point can be added and scalar multiplied.

You’ll need to bone up on the chain rule from calculus 101.

A few more definitions — natural coordinate functions, tangent space to R^3 at p, vector field on R^3, pointwise principle, natural frame field, Euclidean coordinate function (written x_i, where i is in { 1, 2, 3 } ), derivative of a function in direction of vector v (e.g. directional derivative), operation of a vector on a function, tangent vector, 1-form, dual space. I had to write them down to keep them straight as they’re mixed in paragraphs containing explanatory text.

and

at long last,

differential of x_i (written dx_i)

All is not perfect.  On p. 28 you are introduced to the alternation rule

dx_i ^ dx_j = – dx_j ^ dx_i with no justification whatsoever

On p. 30 you are given the formula for the exterior derivative of a one form again with no justification.  So its back to mumbling incantations and plug and chug

Understanding the Riemann curvature tensor is like doing a spinal tap

Back in the day when I was doing spinal taps, I spent far more time setting them up (positioning the patient so that the long axis of the spinal column was parallel to the floor and the vertical axis of the recumbent patient was perpendicular to the floor) than actually doing the tap.  Why? because then, all I had to do was have the needle parallel to the floor, with no guessing about how to angle it when the patient had rolled (usually forward into the less than firm mattress of the hospital bed).

So it is with finally seeing what the Riemann curvature tensor actually is, why it is the way it is, and why the notation describing it is such a mess.  Finally on p. 290 of Needham’s marvelous book “Visual Differential Geometry and Forms” the Riemann curvature tensor is revealed in all its glory.  Understanding it takes far less time than understanding the mathematical (and geometric) scaffolding required to describe it, a la spinal taps.

Over the years while studying relativity, I’ve seen it in one form or other (always algebraic) without understanding what the algebra was really describing.

Needham will get you there, but you have to understand a lot of other things first. Fortunately almost all of them are described visually, so you see what the algebra is trying to describe.  Along the way you will see what the Lie bracket of two vector fields is all about along with holonomy.  And you will really understand what curvature is.  And Needham will give you 3 ways to understand parallel transport (which underlies everything — thanks Ashutosh)

Needham starts off with Gauss’s definition of curvature of a surface — the angular excess of a triangle, divided by its area.

Here is why this definition is enough to show you why the surface of a sphere is curved.   Go to the equator.  Mark point one, then point two 1/4 of the way around the sphere.  Form longitudes (perpendiculars) there and extend them as great circles toward the North pole. You now have a triangle containing 3 right angles, (clearly an angular excess from Euclid who states that the sum the angles of a triangle is two right angles).  The reason, of course, is because the sphere is curved.

Ever since I met a classmate 12 years ago at a college reunion who was a relativist working with Hawking, I decided to try to learn relativity so I’d have something intelligent to say to him if we ever met again (COVID19 stopped all that although we’re still both alive).

Now that I understand what the math of relativity is trying to describe, I may be able to understand general relativity.

Be prepared for a lot of work, but do start with Needham’s book.  Here are some links to other things I’ve written about it.  It will take some getting used to as it is very different from any math book you’ve ever read (except Needham’s other book).

12 July 21 — https://luysii.wordpress.com/2021/07/12/a-premature-book-review-and-a-60-year-history-with-complex-variables-in-4-acts/

4 Dec 21 — https://luysii.wordpress.com/2021/12/04/a-book-worth-buying-for-the-preface-alone-or-how-to-review-a-book-you-havent-read/

7 Mar 22 — https://luysii.wordpress.com/2022/03/07/visual-differential-geometry-and-forms-q-take-3/

27 June 22 — https://luysii.wordpress.com/2022/06/27/the-chinese-room-argument-understanding-math-and-the-imposter-syndrome/

17 July 22 — https://luysii.wordpress.com/2022/07/17/a-visual-proof-of-the-the-theorem-egregium-of-gauss/

Apologies for another posting delay

Hopefully the post on the paper I’m so impressed with will be out in the next few days.  I’ve been clearing away the underbrush in Needham’s Visual Differential Geometry and Forms before the final push on the Einstein field equation and Riemannian geometry.

Apologies for the delay

Here’s a clue for you all to think about — what effects does proline have on (1) the alpha helix (2) the beta pleated sheet?

Bye bye stoichiometry

I’m republishing this old post from 2018, to refresh my memory (and yours) about liquid liquid phase separation before writing a new post on one of the most interesting papers I’ve read in recent years.  The field has exploded since this was written.

Until recently, developments in physics basically followed earlier work by mathematicians Think relativity following Riemannian geometry by 40 years.  However in the past few decades, physicists have developed mathematical concepts before the mathematicians — think mirror symmetry which came out of string theory — https://en.wikipedia.org/wiki/Mirror_symmetry_(string_theory). You may skip the following paragraph, but here is what it meant to mathematics — from a description of a 400+ page book by Amherst College’s own David A. Cox

Mirror symmetry began when theoretical physicists made some astonishing predictions about rational curves on quintic hypersurfaces in four-dimensional projective space. Understanding the mathematics behind these predictions has been a substantial challenge. This book is the first completely comprehensive monograph on mirror symmetry, covering the original observations by the physicists through the most recent progress made to date. Subjects discussed include toric varieties, Hodge theory, Kahler geometry, moduli of stable maps, Calabi-Yau manifolds, quantum cohomology, Gromov-Witten invariants, and the mirror theorem. This title features: numerous examples worked out in detail; an appendix on mathematical physics; an exposition of the algebraic theory of Gromov-Witten invariants and quantum cohomology; and, a proof of the mirror theorem for the quintic threefold.

Similarly, advances in cellular biology have come from chemistry.  Think DNA and protein structure, enzyme analysis.  However, cell biology is now beginning to return the favor and instruct chemistry by giving it new objects to study. Think phase transitions in the cell, liquid liquid phase separation, liquid droplets, and many other names (the field is in flux) as chemists begin to explore them.  Unlike most chemical objects, they are big, or they wouldn’t have been visible microscopically, so they contain many, many more molecules than chemists are used to dealing with.

These objects do not have any sort of definite stiochiometry and are made of RNA and the proteins which bind them (and sometimes DNA).  They go by any number of names (processing bodies, stress granules, nuclear speckles, Cajal bodies, Promyelocytic leukemia bodies, germline P granules.  Recent work has shown that DNA may be compacted similarly using the linker histone [ PNAS vol.  115 pp.11964 – 11969 ’18 ]

The objects are defined essentially by looking at them.  By golly they look like liquid drops, and they fuse and separate just like drops of water.  Once this is done they are analyzed chemically to see what’s in them.  I don’t think theory can predict them now, and they were never predicted a priori as far as I know.

No chemist in their right mind would have made them to study.  For one thing they contain tens to hundreds of different molecules.  Imagine trying to get a grant to see what would happen if you threw that many different RNAs and proteins together in varying concentrations.  Physicists have worked for years on phase transitions (but usually with a single molecule — think water).  So have chemists — think crystallization.

Proteins move in and out of these bodies in seconds.  Proteins found in them do have low complexity of amino acids (mostly made of only a few of the 20), and unlike enzymes, their sequences are intrinsically disordered, so forget the key and lock and induced fit concepts for enzymes.

Are they a new form of matter?  Is there any limit to how big they can be?  Are the pathologic precipitates of neurologic disease (neurofibrillary tangles, senile plaques, Lewy bodies) similar.  There certainly are plenty of distinct proteins in the senile plaque, but they don’t look like liquid droplets.

It’s a fascinating field to study.  Although made of organic molecules, there seems to be little for the organic chemist to say, since the interactions aren’t covalent.  Time for physical chemists and polymer chemists to step up to the plate.

A visual proof of the the theorem egregium of Gauss

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.

and Here — https://luysii.wordpress.com/2022/03/07/visual-differential-geometry-and-forms-q-take-3/

 

 

OK, so what 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
have to even look at an object in 3 space,  just perform local measurements (using the distance between surface points, e.g. the metric e.g.  the metric tensor) .  Curvature is intrinsic to the surface itself, and you don’t have to get outside of the surface (as we are) to find it.

 

 

The idea (and mathematical machinery) has been extended to the 3 dimensional space we live in (something we 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).

 

So how does Callahan phrase and prove the theorem egregium? He defines curvature as the ratio of the area on a (small) patch on the surface to the area of another patch on the unit sphere. If you took some vector calculus, you’ll know that the area spanned by two nonCollinear vectors is the numeric value of their cross product.

 

 

The vectors Callahan needs for the cross product are the normal vectors to the surface.  Herein beginneth the algebra. Callahan parameterizes the surface in 3 space from a region in the plane, uses the metric of the surface to determine a formula for the normal vector to the surface  at a point (which has 3 components  x , y and z,  each of which is the sum of 4 elements, each of which is the product of a second order derivative with a first order derivative of the metric). Forming the cross product of the normal vectors and writing it out is an algebraic nightmare.  At this point you know you are describing something called curvature, but you have no clear conception of what curvature is.  But you have a clear definition in terms of the ratio of areas, which soon disappears in a massive (but necessary) algebraic fandango.

 

 

On pages 258 – 262 Callahan breaks down the proof into 9 steps involving various mathematical functions of the metric and its derivatives such as  Christoffel symbols,  the Riemann curvature tensors etc. etc.  It is logically complete, logically convincing, and shows that all this mathematical machinery arises from the metric (intrinsic to the surface) and its derivatives (some as high as third order).

 

 

For this we all owe Callahan a great debt.  But unfortunately, 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.

 

 

Callahan’s proof is the way Gauss himself did it and Callahan told me that Gauss didn’t have the notational tools we have today making the theorem even more outstanding (egregious).

 

Well now,  onto Needham’s geometrical proof.  Disabuse yourself of the notion that it won’t involve much intellectual work on your part even though it uses the geometric intuition you were born with (the green glasses of Immanuel Kant — http://onemillionpoints.blogspot.com/2009/07/kant-for-dummies-green-sunglasses.html)

 

Needham’s definition of curvature uses angular excess of a triangle.  Angles are measured in radians, which is the ratio of the arc subtended by the angle to the radius of the circle (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.

 

Here is a triangle with angular excess.  Start with a sphere of radius R.  Go to the north pol and drop a longitude down to the equator.  It meets the equator at a right angle (pi/2).  Go back to the north pole, form an angle of pi/2 with the first longitude, and drop another longitude at that angle which meets the equator at an angle of pi/2.   The two points on the equator and the north pole form a triangle, with total internal angles of 3*(pi/2).  In plane geometry we know that the total angles of a triangle is 2 (pi/2).  (Interestingly this depends on the parallel postulate. See if you can figure out why).  So the angular excess of our triangle is pi/2.  Nothing complicated to understand (or visualize) here.

 

Needham defines the curvature of the triangle (and any closed area) as the ratio between the angular excess of the triangle to its area

 

 

What is the area of the triangle?  Well, the volume of a sphere is (4/3) pi * r^3, and its area is the integral (4 pi * r^2).  The area of the north hemisphere, is 2 pi *r^2, and the area of the triangle just made is 1/2 * Pi * r^2.

 

 

So the curvature of the triangle is (pi/2) / (1/2 * pi * r^2) = 1 / r^2.   More to the point, this is the curvature of a sphere of radius r.

 

 

At this point you should have a geometric intuition of just what curvature is, and how to find it.  So when you are embroiled in the algebra in higher dimensions trying to describe curvature there, you will have a mental image of what the algebra is attempting to describe, rather than just the symbols and machinations of the algebra itself (the Lilliputian strands of logic tying down the Gulliver of curvature).

 

The road from here to the Einstein gravitational field equations (p. 326 of Needham) and one I haven’t so far traversed,  presently is about 50 pages.Just to get to this point however,  you have been exposed to 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.

The Chinese Room Argument, Understanding Math and the imposter syndrome

The Chinese Room Argument

 was first published in a 1980 article by American philosopher John Searle. He imagines himself alone in a room following a computer program for responding to Chinese characters slipped under the door. Searle understands nothing of Chinese, and yet, by following the program for manipulating symbols and numerals just as a computer does, he sends appropriate strings of Chinese characters back out under the door, and this leads those outside to mistakenly suppose there is a Chinese speaker in the room.

 

So it was with me and math as an undergraduate due to a history dating back to age 10.  I hit college being very good at manipulating symbols whose meaning I was never given to understand.  I grew up 45 miles from the nearest synagogue.  My fanatically religious grandfather thought it was better not to attend services at all than to drive up there on the Sabbath.  My father was a young lawyer building a practice, and couldn’t close his office on Friday.   So my he taught me how to read Hebrew letters and reproduce how they sound, so I could read from the Torah at my Bar Mitzvah (which I did comprehending nothing).  Since I’m musical, learning the cantillations under the letters wasn’t a problem.

 

I’ve always loved math and solving problems of the plug and chug variety was no problem.  I’d become adept years earlier at this type of thing thanks to my religiously rigid grandfather.   It was the imposter syndrome writ large.  I’ve never felt like this about organic chemistry and it made a good deal of intuitive sense the first time I ran into it.  For why have a look at — https://luysii.wordpress.com/2012/09/11/why-math-is-hard-for-me-and-organic-chemistry-is-easy/

 

If there is anything in math full of arcane symbols calling for lots of mechanical manipulation, it is the differential geometry and tensors needed to understand General relativity.   So I’ve plowed through a lot of it, but still don’t see what’s really going on.

 

Enter Tristan Needham’s book “Visual Differential Geometry and Forms”.  I’ve written about it several times
and Here — https://luysii.wordpress.com/2022/03/07/visual-differential-geometry-and-forms-q-take-3/

 

If you’ve studied any math, his approach will take getting used to as it’s purely visual and very UNalgebraic.  But what is curvature but a geometric concept.

 

So at present I’m about 80 pages away from completing Needham’s discussion of general relativity.  I now have an intuitive understanding of curvature, torsion, holonomy, geodesics and the Gauss map that I never had before.   It is very slow going, but very clear.  Hopefully I’ll make it to p. 333.  Wish me luck.

Visual Differential Geometry and Forms — Take 3

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

here — https://luysii.wordpress.com/2021/07/12/a-premature-book-review-and-a-60-year-history-with-complex-variables-in-4-acts/

and

here — https://luysii.wordpress.com/2021/12/04/a-book-worth-buying-for-the-preface-alone-or-how-to-review-a-book-you-havent-read/

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 — https://en.wikipedia.org/wiki/Occipital_lobe 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.

A book worth buying for the preface alone (or how to review a book you haven’t read)’

Everyone in grade school writes a book report on something they haven’t completely read (or read at all). Well, I’m long past, but here’s a book worth buying for the preface alone — Visual Differential Geometry and Forms by Tristan Needham.

His earlier book Visual Complex Analysis (another book I haven’t read completely) is a masterpiece (those parts I’ve managed to read) with all sorts of complex formulas and algebraicism explained visually.  It’s best to be patient, as Needham doesn’t even get to complex differentiation until Chapter 4 (p. 188) The Amplitwist concept.

On page xviii of the preface, Needham describes the third type of calculus Newton invented, and the one he used in the great Principia Mathematica of 1687.

The first one of 1665 was basically the manipulation of power series (which I’ve never heard of.  Have you?)

The second was one we all study in school, dx dy and all that.

Newton called the third method of calculus “the synthetic method of fluxions.”

Bishop Berkeley had a field day with them.  “And what are these fluxions? The velocities of evanescent increments? And what are these same evanescent increments? They are neither finite quantities, nor quantities infinitely small, nor yet nothing. May we not call them ghosts of departed quantities?”

Laugh not at the good Bishop;   Berkeley California is named for him (as long as the woke don’t find out).

Needham gives a completely nontrivial example of the method.  He shows geometrically how the derivative of tan(theta) is 1 + [ tan(theta)]^2,.

The diagram will not be reproduced here but suffice it to say you’ll have to remember a lot of your geometry, e.g. two lines separated by a very small angle are essentially parallel, the radius of a circle is perpendicular to the tangent, what a similar triangle is.

The best thing about it is you can actually visualize the limit process taking place as a triangle shrinks and shrinks and eventually becomes ‘almost’ similar to another.  No deltas and epsilons for Newton (or Needham).  There is some algebra involving the ratios of the sides of similar triangles, but it’s trivial.

Buy the book and enjoy.   You’ll never think of differentiation the same way.  Your life will be better, and you might even meet a better class of people.

Fiber bundles at last

As an  undergraduate, I loved looking at math books in the U-store.  They had a wall of them back then, now it’s mostly swag.  The title of one book by a local prof threw me — The Topology of Fiber Bundles.

Decades later I found that to understand serious physics you had to understand fiber bundles.

It was easy enough to memorize the definition, but I had no concept what they really were until I got to page 387 of Roger Penrose’s marvelous book “The Road to Reality”.  It’s certainly not a book to learn physics from for the first time.  But if you have some background (say just from reading physics popularizations), it will make things much clearer, and will (usually) give you a different an deeper perspective on it.

Consider a long picket fence.  Each fencepost is just like every other, but different, because each has its own place.  The pickets are the fibers and the line in the ground on which they sit is something called the base space.

What does that have to do with our 3 dimensional world and its time?

Everything.

So you’re sitting at your computer looking at this post.  Nothing changes position as you do so.  The space between you and the screen  is the same.

But the 3 dimensional space you’re sitting in is different at every moment, just as the pickets are different at every position on the fence line.

Why?  Because you’re siting on earth.  The earth is rotating, the solar system is rotating about the galactic center, which is itself moving toward the center of the local galactic cluster.

Penrose shows that this is exactly the type of space implied by Galilean relativity. (Yes Galileo conceived of relativity long before Einstein).   Best to let him speak for himself.   It’s a long quote but worth reading.

“Shut yourself up with some friend in the main cabin below decks on some large ship, and have with you there some flies, butterflies, and other small flying animals. Have a large bowl of water with some fish in it; hang up a bottle that empties drop by drop into a wide vessel beneath it. With the ship standing still, observe carefully how the little animals fly with equal speed to all sides of the cabin. The fish swim indifferently in all directions; the drops fall into the vessel beneath; and, in throwing something to your friend, you need throw it no more strongly in one direction than another, the distances being equal; jumping with your feet together, you pass equal spaces in every direction. When you have observed all these things carefully (though doubtless when the ship is standing still everything must happen in this way), have the ship proceed with any speed you like, so long as the motion is uniform and not fluctuating this way and that. You will discover not the least change in all the effects named, nor could you tell from any of them whether the ship was moving or standing still. In jumping, you will pass on the floor the same spaces as before, nor will you make larger jumps toward the stern than toward the prow even though the ship is moving quite rapidly, despite the fact that during the time that you are in the air the floor under you will be going in a direction opposite to your jump. In throwing something to your companion, you will need no more force to get it to him whether he is in the direction of the bow or the stern, with yourself situated opposite. The droplets will fall as before into the vessel beneath without dropping toward the stern, although while the drops are in the air the ship runs many spans. The fish in their water will swim toward the front of their bowl with no more effort than toward the back, and will go with equal ease to bait placed anywhere around the edges of the bowl. Finally the butterflies and flies will continue their flights indifferently toward every side, nor will it ever happen that they are concentrated toward the stern, as if tired out from keeping up with the course of the ship, from which they will have been separated during long intervals by keeping themselves in the air. And if smoke is made by burning some incense, it will be seen going up in the form of a little cloud, remaining still and moving no more toward one side than the other. The cause of all these correspondences of effects is the fact that the ship’s motion is common to all the things contained in it, and to the air also. That is why I said you should be below decks; for if this took place above in the open air, which would not follow the course of the ship, more or less noticeable differences would be seen in some of the effects noted.”

I’d read this many times, but Penrose’s discussion draws out what Galileo is implying. “Clearly we should take Galileo seriously.  There is no meaning to be attached to notion that any particular point in space a minute from now is to be judged as the same point in space that I have chosen. In Galilean dynamics we do not have just one Euclidean 3-space as an arena for the actions of the physical world evolving with time, we have a different E^3 for each moment in time, with no natural identification between these various E^3 ‘s.”

Although it was obvious to us that the points of our space retain their identity from one moment to the next, they don’t.

Penrose’s book is full of wonderful stuff like this.  However, all is not perfect.  Physics Nobelist Frank Wilczek in his review of the book [ Science vol. 307 pp. 852 – 853 notes that “The worst parts of the book are the chapters on high energy physics and quantum field theory, which in spite of their brevity contain several serious blunders.”

However, all the math is fine, and Wilczek says “the discussions of the conformal geometry of special relativity and of spinors are real gems.”

Since he doesn’t even get to quantum mechanics until p. 493 (of 1049) there is a lot to chew on (without worrying about anything other than the capability of your intellect).