A Note on Teaching Math, Unexpected Art, and Picturing Things Differently

JF Ptak Science Books   Post  1427

Stereo viewing geometrical figures may or may not be useful--I find it a little off-putting, creating a slight dizzying effect, though I'm sure it helps others (particularly children?) see complex geometrical figures a little more easily., especially when it comes to crystalography images. Education was certainly what Henry Vuibert , the author of Les Anaglyphes Geometriques had in mind when he published the work in 1912.  He had a long history in mathematics education, having founded the firm that bears his name back in 1877, a publishing house specializing in school texts in the maths, physics and general sciences.  (We offer the book for purchase at out blog bookstore.)

Vuibert's book is a very early example of the use of the anaglyphe--examples of the use of the method go back to the 1850's, when d'Almeida used a version of the idea to punctuate lectures using a magic lantern; the (printing) process of the anaglyphe wasn't patented until the 1890's by L.D. DuHaron.  It does confuse me or remind me quite a bit of Oliver Byrne's beautiful monster The First Six Books of Euclid in which coloured diagrams and symbols are used instead of letters for the greater ease of readers... printed at the Chiswick Press by C. Whittingham for William Pickering in 1847.  The book is, well, unusual and probably not at all usefu1¹.  Bryne replaced all of the algebraic notation, identifying letters and almost all of the descriptive text with color and color codes, leaving Euclid mysterious, hidden, awkward, impervious, and, yes, beautiful². As a matter of opinion this work presages the cubists (and especially Piet Mondrian) by 60 years, and all by accident artwork. 

 

 

 

 

 

 

 

 

 

 

 

There are certainly much earlier works that tried to get at a fuller representation of geometrical objects, not the least of which was a 16th century Elements of Euclid in which the figures were cut-out paper, pasted into the book and attached to a string so that the reader could pull out the illustration and see it displayed in three dimensions. The elements of geometrie of the most auncient philosopher Euclide of Megara. Faithfully (now first translated into the Englishe toung…³, printed in London in 1570, was no doubt a great and exasperating labor of love, a complex "pop-up" book accomplished with sixteenth century technology.  (The photo below shows the Smithsonian Institution's copy.)

Of course there's the very old fashioned way of representing mathematical concepts, and that's with sticks and stones and blocks and such.  For visceral memory and practical applications, the idea is hard to beat, unless of course, you really beat it , which is difficult to do.  But to me that's just what Major G. Mackinlay did in his Realistic Arithmetic,  for Teaching Children the Elements of Arithmetic by Means of Special Blocks⁴, published in London in 1900--his ideas for the use of blocks was very sound and stable; his instructions (and in particular his illustrations) for what to do with the blocks was not.  I am reminded of inscrutable, badly translated and designed instructional manuals for tech stuff that need a manual for the manual. 

Here's one of the folding plates describing how to assemble the blocks:

Then there are other sorts of wooden (and etc.) three-dimensional models, such as the (600+) gorgeous geometric designs housed in the Institute Henri Poincare, transferred there from the laboratory of geometry of the Faculty of Sciences of the University of Paris in 1928⁵.  Most of these models were constructed around the time of vast change in mathematics and art (1900-1920), which means that many of them predate the revolutionary ascensions of  Cubism and non-representational art.

And as long as were at the point of unintentional and under-recogniozed epochal works, I think that there is a broad book that could be done on exactly this topic, calling into play such examples as Georgina Houghton, Victor Higo, Emily Vanderpoel, and many others. 

Georgina Houghton, for example, produced this non-representational painting in 1870 (or thereabouts):

 And Victor Hugo, from the 1860's

And Emily Vanderpoel (1900 or so):

In another, much earlier case of removing detail and adding abstraction for the sake of simplicity in representing an idea,  Erhard Schoen, in his Unnderweissung der proportzion und stellung der posssen liegent und dtehent, printed in Nurenberg  in 1538, presented these cubed figures (below).  He used simplified geometric form to stand in for the curvy humans, replacing them with proportional stacks of boxes which would more easily explain to the younger reader how to represent the human body in space and in proportion to other things. 

I think that these were monumentally combative images showing humans in a radical, previously unknown way, much like the moving-through-space-and-time photographs of the somewhat forgotten Etienne Marey, who in the 1870's created what was essentially the world's first "slow motion" device. One iteration of Marey's apparatus was basically a long series of ganged cameras recording a motion for a simple task at a given time frame and presented on a continuous strip of photographic paper, sort of like a motion picture with the camera speed set at three frames per second. The resulting images were phenomenal and showed people for the first time the exactness of all manners of simple motions--motions that no longer looked so "simple" once all of its aspects could be studied from captured photographic evidence.  Even the act of hopping over a small stool or bending to pick up a bucket of water were enormously revealing in a way like Robert Hooke's Micrographia displayed the great detail and complexity of the seemingly simple fly.  Perhaps the most famous of Marey's series of images was that of a galloping horse, which also for the first time revealed what exactly the horse's legs were doing and proving that almost every painter in the history of art represented the galloping horse incorrectly.  His series of photographs (as in this sample below) show a fairly close fit to the work of the futurists (like for example Duchamp) who would come into being after another four decades. 

  And Duchamp's Nude Descending (1912):

 

I do want to make it clear that this is just a simple note, a thinking-out-loud piece on how attempts to visualize things differently for the sake of explanation or education can come "unintentionally" close to what would become epochal works.  More later.

Notes

1. Augustus De Morgan, mathematician and logician, wrote a very highly critical book A Budget of Paradoxes in which he describes hubbub, fakirs, frauds, perpetual motion machines, squaring the circle efforts, millstones and other useless books in the maths, and in here he sniffily dismisses Byrne's work. At best, to De Morgan, the Byrne book is "curious". But useless and curious, as I have seen many thousands of times, does not mean that it can't be attractive and beautiful, and the Byrne book is probably the leading candidate in the Bad & Beautiful category.

2. In his Victorian Book Design McLean calls the Byrne book "...one of the oddest and most beautiful books of the whole century...a decided complication of Euclid, but a triumph for Charles Whittingham [the printer]".

3.  This is also the first appearance of Euclid in English.  The full title: The Elements of geometrie of the most auncient philosopher Euclide of Megara faithfully (now first) translated into the Englishe toung by H. Billingsley, citizen of London; whereunto are annexed certaine scholies, annotations, and inventions, of the best mathematiciens, both of time past and in this our age; with a very fruitfull praeface made by M.I. Dee, specifying the chief mathematicall scieces, what they are, and whereunto commodious; where, also are disclosed certaine new secrets mathematicall and mechanicall, untill these our daies, greatly missed. London, Imprinted by I. Daye, 1570.

4.  From the Report of the Annual Meeting of the British Association for the Advancement of Science, 1906:

 

5.  From the IHP website:  "Most of the models  were created by Martin Schilling in Leipzig between 1900 and 1920 and  a few of them  (in wood)  were made between 1912 and 1915 by J. Caron, Professor of Descriptive Geometry in charge of  the "graphic work" course at the ENS at the beginning of the century (between 1912 and 1915)."

Also this:   "In the thirties, the surrealists, led by Max Ernst, were interested in these geometric objects. André Breton alludes to them in  "Crise de l'objet" , an article  in the magazine  Cahiers d'Art (May 1936, No. 1-2, p. 21-26.). Man Ray took photographs of the models, which where published in same issue. He would later use them to compose what he called  "Shakespearean Equations". Many artists, painters, architects and sculptors drew inspiration from them."