More History of Maths

Professor A.W.F. Edwards: Venn diagrams

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As with most capitalized innovations, diagrams with overlapping areas did not originate with the person they’re named after; Venn was adapting diagrams made by Euler, who was likely adapting his from elsewhere. The BSHM bulletin even posted an 11th century overlapping music diagram deriving from someone named John of Afflighem. Euler (and likely John) were depicting the results of a logical structure already known. Venn, instead, used the diagrams to help discover what the structure is.

But the talk focused on Venn diagrams and how many overlapping areas they can have. The more areas you have, the more their shape has to change. Four areas become elliptical, and more tend to become sausage-shaped and curved.

At that point, it seemed to me that the actual information was being lost in complexity. But Professor Edwards’ field is genetics, and he showed how mapping the genetic code for amino acids could lead to new discoveries. Even understanding nothing about Mendelian biology or genetics, I was able to see the usefulness of the creation of adjoining areas on the resulting Venn diagram, which suggested overlaps and duplications in the code. It was clear that new and important scientific discoveries could be made just by changing the visualization of data.

Professor June Barrow-Green: Olaus Henrici

Henrici was supposed to become an engineer, but went into maths instead. Ferdinand Redtenbacher, one of his professors in Mechanical Engineering at Karlruhe Polytechnium, had created over 100 models for teaching engineering (again, we have innovation as a result of teaching students).

Although Barrow-Green emphasized Henrici’s move to pure mathematics, he never went far from what I would call applied maths. She discovered a patent of his for an improvement in the construction of bridges, arches, and roofs, and he published a book called Skeleton Structures, which, although it made him no money, became foundational reading for the building of skyscrapers in the United States.

As with the story of H.G. Wells (sorry), Henrici’s problem was figuring out how to make a living from his varied talents. And, like H.G., he became a tutor to make ends meet. His opportunity came at Central Technical College (now part of Imperial College, so part of the South Kensington mode in which H.G. trained). There he was able to create a Mechanics Laboratory in 1884 (I really have to wonder whether he and Wells ever ran into each other).

His central interest since working with Clebsch as a young man, however, had been pure maths, and in particular geometry (not surprising given his interest in construction). He began to develop his own system for teaching a “modern” geometry, finding the emphasis on Euclid to be underwhelming in preparing students. Teachers, apparently, knew this, and yet the university curriculum, even at Cambridge, was still teaching Euclid almost exclusively. Henrici created cardboard and stick models to teach his modern geometry, and wrote a textbook (as I’m starting to think all good teachers do).

Apparently Charles Dodgson (his pen name was never mentioned or implied) wrote a thesis on “Euclid and His Modern Rivals” discussing the new methods. I need to explore which kind of geometry Wells was exposed to for his exams.

Although Henrici didn’t publish many research papers (teaching, remember?), he joined the Royal Society and became famous for the mathematical models, which influenced others.

Professor Sarah Hart: Symmetry, Pattern and Groups

Perhaps symmetry makes me sleepy, or maybe it was because the session was after lunch, but my notes say, “your major should be the topic wherein you wouldn’t fall asleep at a good lecture, even if you were tired”.

Image result for repeating pattern ancient griffin friezeWhile starting with images, this talk spent much time on formulas for what I would call repetitive elements of design, and I’m afraid I never understood the difference between a reflection and other ways that elements repeat. We began with Platonic solids and ended with Coxeter graphs, and the only part that made sense to me actually came from a question from another mathematician. Symmetry, among other things, makes it possible to reduce the number of instructions, because elements repeat. I see the connection here to understanding the natural world, which has many symmetries (a word which more me means many examples of symmetry, but for Professor Hart means something else entirely). I understand the desire to reduce the set of instructions so as to make more complex findings. This gelled with my understanding of the purpose of mathematics: to reduce the set of instructions to its simplist forms.

Kenneth Falconer: Fractals – Simple or Complex?

Not even sure what a fractal is, I paid close attention. Apparently calculus in the 18th and 19th century was applicable only to smooth curves, which could have a tangent. But something called the von Koch curve is so irregular that you cannot draw a tangent anywhere. (This made me thing of student complaints that professors’ lectures “go off on a tangent” – perhaps if our lecture is super-irregular there would be not tangents!).

However, somewhere the curve is “self-similar”, which seems to mean repeating in some way, so I had trouble meshing this with my understanding of “irregular”, which to me means without regularities. Image:Sierpinski clear.gifExamples like the Sierpinski triangle looked regular to me (and symmetrical).

Benoit Mandelbrot claimed that irregular shapes are the norm (examples today might be fluid dynamics, the branching of airways in the lungs, the stock market patterns), which meant that fractals are the norm, which meant that mathematics to study them needed to be developed. Here we got into a competition to do this after the First World War, and the necessary win by Gston Julia because he was a war hero, even though Pierre Fatou (the only other competitor) submitted work that was as good.

Image result for julia fractals

If a Mandelbrot set starts at 0,0 and uses the formula, it will create a cohesive shape (that looks like a strange, but symmetrical, sea creature), but if it starts in some other places, the coordinates become “dust-like” rather than connected. With the advent of computer modeling beginning in the 70s, it became possible to calculate these out to the points where further miniature Mandelbrot sets could be seen in the connected portions.

I was unable to determine the significance of all this, and it still looked fairly regular to me, and seemed rather self-referential. But, as my son has told me, mathematics is circular anyway.

Ian Stewart: Picturing Chaos

The thesis here was that it was pictures that drove the research of chaos, rather than the mathematics leading to the pictures, and here we really did get into irregularities. I knew nothing about chaos theory going in (it always seemed to me like the word “anarchy” – something that has a specific definition to scholars, but that lay people interpret wrongly).

Newton’s physics could explain the interaction of two heavenly bodies, but once you added a third the mathematics fell apart. Image result for poincare homoclinic tangleHenri Poincare tried it with two bodies with much mass and one with little mass, and it created such complexity that he said he could not draw the result. He concluded that Newton’s method simply wouldn’t work to picture the impact of three bodies upon each other – the curves fold back on each other, and make what’s called a Homoclinic Tangle.

Enter Edward Lorenze, who in 1961 was using a room-sized computer to do calculations, and turned it off in the middle of a process because he didn’t want to leave it running while he was gone. When he returned, he reset the machine back a few calculations, to make sure he’d re-entered everything correctly, and he noted the next several calculations were correct and matched. So he let it run, but after awhile he noted the results were not the same at all.

Technicians told him this was because the computer stored the numbers to many more decimal points than could be entered on the keys, but that it was a very tiny change. With ongoing calculations, however, the change got bigger and bigger. Although it may have been a computer bug (or feature!) to begin with, the implications were bigger. It mean that one tiny, inperceptible change someone could change the entire outcome. More, the results will ultimately diverge until they are on entirely different, unrelated trajectories. He published his paper on this (Deterministic Nonperiodic Flow), but it was ignored by his intended audience of meteorologists and by mathematicians. When he would later present on it, but hadn’t given the talk a title, the conference organizer called it “The Butterfly Effect”.

The theory was proven in 2002 with computers, and is now regularly used to predict weather. This was Lorenze’s original research, to determine how to better predict the weather. By showing the unexpected results of even a tiny change, he changed meteorology into something that requires far more predictive models, not to create certainty but to provide a range of possible scenarios which can then be combined into the most likely one. The implications, of course, are much larger – one question got into catastrophic theory, and how the issuing of multiple possible results could create better systems for monitoring bodily processes like organ function.

History of Mathematics class

We pause in the narrative of wonderful Wellsian things for a brain shift: a class from Oxford University Deapartment for Continuing Education offered by the British Society for the History of Mathematics. I had been to a similar class of theirs last year, in the same location at Rewley House, and really enjoyed it. Although somewhat like American extension classes, these are at a higher level, more like a conference. The room is full of Society members, mathematicians, and historians (note use of Oxford comma there). Last year I met the curator of the History of Science museum, for example.

The topic/title was “Picturing Mathematics”, but of course it’s really about history, so all papers were presented in chronological order. The class began with some issues (no coffee, no soap in the ladies’, computer problems) but really was fine. Some Notes, Part I:


Professor Jeanne Peiffer: The painter’s eyes in Albrecht Dürer’s geometrical diagram

An analysis of Dürer’s diagrams and methods from his Manual of Measurement (1525), focusing on the combination of practical use (for artists and craftsmen) and mathematics. For me, the most interesting aspect was that the book as “written for the young and for those who lack a devoted instructor”, those who have nobody to train them. Dürer said you didn’t even need his book if you already knew Euclid. Although not a theme of the conference, I would notice throughout the day how many marvelous diagrams and images have been created in order to teach them to students.

As part of his textbook (it really is a textbook) Dürer wanted the images in juxtaposition to explanatory text. This I have previously blogged about as being terribly important in online learning – proximity of image to text is crucial. In his introduction, Dürer wrote that the text and images must correspond because “the inner understanding is demonstrated by external exposition”. He wrote, “[a]nd for this reason I shall draw all things described in this book next to their explanations…” Nice to be in agreement with an artist and scholar from the 16th century. It’s the historian’s creed: Nothing is New.

Professor Hugh Small: Florence Nightingale’s statistical diagrams

This was the topic that first attracted me to the class, since I have a strong interest in the history of medicine during the Victorian era (and am considering writing a mystery novel based on that interest). Professor Small’s book is now on my must-buy list. A primary focus was on the development and use of this graphic, which Florence Nightingale created in her push for sanitation reform:

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The first fascinating thing about the chart is that today we interpret it wrongly. Although it certainly shows that the amount of deaths due to disease during the Crimean War vastly exceeded the number of deaths due to war wounds, this was not the point Nightingale was trying to make. No Victorians, apparently, would have been surprised by this — death by disease in war-time had always exceeded deaths in battle.

Now that I know Professor Small is also trained in psychology, I understand how he was able to so clearly explain how graphics are interpreted: we see what surprises us. This is apparently because our visual acumen has developed for much longer in humans than our verbal acumen. When we hear something, we tend to listen for what we expect, but visually we are trained to look for things that are out of place.

So what Nightingale was trying to do, and what Victorian people would have seen, was the decline in death by disease when comparing the first (larger image) and second years of the war. What she was showing was that disease could be reduced and prevented. This was important because the elite medical community, which dominated policy in the Privy Council, had been demanding more public support to make scientific discoveries, while the trend toward public sanitation (championed by Edwin Chadwick in the 1840s) had gone out of fashion (the main board for public health had been dismantled in the 1850s). Nightingale, for example, campaigned for landlords to have to pay to hook up all houses to the new sewer systems in London.

The design of the chart is important . The statistics could just have been listed, but the spiral makes you focus differently on the information. Here it is as a bar chart:

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Professor Small pointed out that it’s hard to see the difference in years, and the blue deaths are overemphasized – it looks like a chart showing things were bad that first winter (which they were, but that wasn’t her point).

She was good at this. Another of her charts he sees as “poetic” (and possible written by a poet), repeating information to get the point across that so many soldiers were dying of preventable diseases right there in the UK due to poor conditions in the barracks:

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The deaths are divided by age for no particular reason – the percentage is about the same in each. But the comparison of deaths of “Englishmen” in general to “English Soldiers”, repeated over and over, hammers home the point.

In 1867, the Reform Act would vastly expand the franchise, and would end the domination of landed interests and the elite medical community, making many of her goals possible.

A bit of William Morris and more Bodleian

My appointment at the Oxford Student Union Library was in the morning yesterday, so I arrived at the iron gates at the appointed time and rang the buzzer. Sue the librarian answered and buzzed me in, then met me in the hall. The building is stunningly beautiful on the outside, at least for those who like Victorian architecture which, as it happens, I do.

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I’m not sure why, but I had not expected the inside to be stunning too. Sue handed me the volumes of H.G. Wells’ Text-book in Biology, and said I could look at them in the library room (they buzz you in there too). For a few minutes I was almost too distracted to look at the book.

0623170933aBut the volumes were wonderful, and show why it’s a bad idea to use online translations to examine books. I have a Project Gutenburg version of these books, but they gave me no sense of scale. The books are wonderfully small – they can be carried in a jacket pocket or bag. Also, the illustrations fold out, which you cannot tell online. Plus, unlike online versions, all the ads are in the back of the actual book – wonderful ads for the University Correspondence College, which I’d mentioned to Sue on the phone I was hoping would be there.  I was so enchanted that I completely missed something. See it at the top of my photo above?

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Well, I didn’t see it. So having photographed pages from the textbooks, including adding some items for scale (see right), I went out to return them and asked about the pre-Raphaelite murals. Was it possible to view them during the school holiday? She gave me the hours, which included the hour I was there, so I asked if I could see them now. Yes, but it’s £1.50, she said apologetically. I paid at the desk and asked where the murals were. How they kept a straight face I don’t know, but they pointed me back into the room where I had just been. The woman at the desk told me that they were hard to see from the ground, and to go up the stairs to the gallery. I laughed and said scholars are so busy looking down, we don’t see what’s around us!

117408870f2f1bdd63968943dcff7ea7The murals are actually very much darkened, but the ceiling is wonderful, because William Morris did it wrong the first time and had to re-do it. However, they had asked that I not photograph (oops, didn’t mean to), so here is someone else’s photo off Pinterest… ->

With no more excuse for being in this fabulous building, I said goodbye, and returned to the Bodleian at 11, when my Educational Times journal came in. I felt a fool as I looked through it and realized I’d committed the classic scholar’s mistake: getting the citation wrong. What the article I’d found had said was that an interview with William Briggs was in the journal Education, not Educational Times, September 1890. It took me quite awhile to find that journal in the online catalog, since the name was so common. Knowing I’d have no time to return the next day, I requested it as “scan and deliver” and hoped for the best (if you request early in the day, sometimes you can get it emailed to you same day).

Realizing this scan and deliver thing was going to be invaluable, and wanting to continue it from the U.S., I went back to the Weston Library and asked to extend the term of my Readers Card. At first they asked me to fill out an application again (it’s very long), but I said I’d gotten it just a few days ago. We both laughed when we realized I’d gotten it the day before. I explained I’m pretty tired from working so hard. Plus London and Oxford have both been in a heat wave that just broke, so everyone is cross. I got my term extended for a year, then at 4 pm the article came in. Another disappointment – the article was the exact same interview that Briggs himself reprinted in the Prospectus I saw at the British Library. Oh well….