The Assignment

In Spring 2012, when I was teaching our upper-division Topics in the History of Mathematics course, I wanted students to interact somehow with primary sources.  This point was driven home when a guest lecturer from the history department explained what primary sources were and their importance to the discipline of history.  Also, I wanted students to do more reading and writing about mathematics (in addition to solving historical math problems), along the lines of what Klyve, Stemkoski, and Tou described  in their comprehensive overview for Convergence of several different ways to use primary sources (and in particular the Euler Archive) in teaching and learning, "Teaching and Research with Original Sources from the Euler Archive."

The research paper assignment that I crafted was admittedly somewhat terse and vague in its wording: “To engage with primary documents (or translations thereof) to answer a math history question.”  Since the class is usually composed of only ten to fifteen math majors and minors, my plan was to meet with each student individually and facilitate a choice of primary source material that appealed to her or his interests.  Then together we would tailor a relevant question (mathematical or historical or both) for the student to answer.  That way, I could point each student toward a problem that would be challenging, but still achievable.

In addition, each student had to choose a source that was different from those selected by other students in the class, which I hoped would lead to more personal investment in their topics, and would avoid the possibility of students working on the same topic and copying each other’s work.  Students could choose sources that we had studied or mentioned in class, they could peruse source books or journals in the library, or they could search online.  The assignment was worth 20% of the course grade, with an expected length of about six pages.  Though it was announced on the syllabus on the first day of class, topic selections were not finalized until after spring break, which meant that students had about one month to complete the assignment. 

While most students did a good job reporting on their primary sources, there was less success with the second part of the assignment: engaging with the source material.  Most of the students who were otherwise performing well in the course simply explained the mathematics of their primary document and did not attempt to extend the mathematics or draw connections to modern formulations of the same topic.  Primary sources selected by these students included some of the more famous documents:

• Leibniz’ proof of the Fundamental Theorem of Calculus;
• Germain’s correspondence with Gauss;
• Euler’s paper on the Königsberg bridges;
• Nash’s dissertation on game theory; and
• various authors’ work on Fermat’s Last Theorem, starting with Fermat’s own infamous comment in the margin of his copy of Bachet’s translation of Diophantus. 

These students wrote interesting reports, but really did not formulate and answer a mathematical or historical question of their own, and so I call this the “report” group.  To be clear, the reports were often well written and novel, but strictly speaking, they did not engage with the material in the way I had tried to outline in the assignment.

Interestingly, the students who were more likely to extend beyond the primary source material – those comprising what I call the "engaged" group – were also more likely to struggle with the other aspects of the course.  Like the students in the "report" group, these students selected some famous works.

• One student chose the Babylonian tablet Plimpton 322, and tried to link it to ancient architecture and fractions, but the paper suffered from a lack of consideration of scholarship on the tablet, such as Eleanor Robson’s “Words and Pictures: New Light on Plimpton 322.”
• Another student wrote about Lagrange’s explanation of Lagrange multipliers in his Mécanique Analytique and created some examples applying Lagrange’s techniques. 
• The author of a paper on Gauss’ Disquisitiones Arithmeticae demonstrated how modular arithmetic could be applied to cryptography.
• And one paper on magic squares, both the early Chinese Lo Shu square and the work of Seki Takakazu (Seki Kowa) in 17th-century Japan, included a derivation of the formula for the sum of any row or column in a normal magic square of order n.

This article will examine in more detail two representative student papers not yet mentioned, one from each group, which deal with lesser-known mathematical documents.  The “report” paper describes Turing’s unpublished work on phyllotaxis, while the “engaged” paper discusses an original number theory article by Euler.  Comparing these two examples of student work will effectively serve as an assessment of the assignment overall, and will point out several ways to improve this assignment in the future.

Turing and Plant Phyllotaxis

This paper is actually a draft, written after Turing’s death by N. E. Hoskin and B. Richards, based on Turing’s original typescript and lecture notes, and was intended to be the first part of a longer paper submitted to the Philosophical Transactions of the Royal Society of London. (Indeed, in a 1952 article for the Transactions, Turing mentioned that he was working on such a paper.)  However, the entire completed paper did not appear in print until 1992, when it was published in Morphogenesis, one of four volumes published by Elsevier that were devoted to Turing’s “mature scientific writings” – as his friend and executor P. N. Furbank wrote in the preface – and even included some unpublished work.  It has recently appeared again in Alan Turing: His Work and Impact, also published by Elsevier, in 2013.  Despite the document not having been written by Turing himself, and despite heavy editing by the actual authors, there is agreement that the ideas contained in it were Turing’s and so I will refer to it as his paper from here on.  For more detail on the document itself, see Jonathan Swinton’s article “Watching the Daisies Grow.”

In the draft version, Turing modeled a portion of a pine branch as a cylinder, and studied the lattice of points where the leaves grew out of the branch.  Probably due to its draft nature, the document contains no figures, not even Figure 1, to which it refers in the passage below [page 2].

The pattern is remarkably regular and is seen to have the following properties.
        1)  If the cylinder is rotated and at the same time shifted along its length in such way as to make a leaf A move into the position previously occupied by a leaf B, every other leaf also moves into a position previously occupied by a leaf.  This may be called the congruence property.
        2)  All the leaves lie at equal intervals along a helix.  On the specimen in Fig. 1 the pitch of the helix is about 0.046 cm and the successive leaves differ in angular position by about 137°.

Turing pointed out that these properties are not independent; clearly any specimen with property 2 also has property 1, but that there are examples in nature of plants having property 1 but not property 2.  Using cylindrical coordinates \((\theta, z)\) on a cylinder of fixed radius, he discussed the idea of parastichy – the helix mentioned above that joins one leaf to the next (and which is very apparent in pineapples and pine cones).  He both counted the number of leaves and measured the central angle between adjacent leaves on each parastichy.  Turing’s Table 1 demonstrates the connection of the Fibonacci numbers to an angle measuring approximately 137.5°.

	Table 1
	DIVERGENCE	ANGLES
Fraction of \(2\pi\)	Degrees, min., sec.	Degrees
\(2\pi\,\dot\,\frac{1}{2}\)	\(180^{\circ}\)
\(2\pi\,\dot\,\frac{1}{3}\)	\(120^{\circ}\)
\(2\pi\,\dot\,\frac{2}{5}\)	\(144^{\circ}\)
\(2\pi\,\dot\,\frac{3}{8}\)	\(135^{\circ}\)
\(2\pi\,\dot\,\frac{5}{13}\)	\(138^{\circ}\,27^{\prime}\,41.5^{\prime\prime}\)	\(138.46154\)
\(2\pi\,\dot\,\frac{8}{21}\)	\(137^{\circ}\,\,8^{\prime}\,34.3^{\prime\prime}\)	\(137.14286\)
\(2\pi\,\dot\,\frac{13}{34}\)	\(137^{\circ}\,38^{\prime}\,49.4^{\prime\prime}\)	\(137.64706\)
\(2\pi\,\dot\,\frac{21}{55}\)	\(137^{\circ}\,27^{\prime}\,16.4^{\prime\prime}\)	\(137.45454\)
\(2\pi\,\dot\,\frac{34}{89}\)	\(137^{\circ}\,31^{\prime}\,41.1^{\prime\prime}\)	\(137.52809\)
\(2\pi\,\dot\,\frac{55}{144}\)	\(137^{\circ}\,30^{\prime}\,00.0^{\prime\prime}\)	\(137.50000\)
\(2\pi\,\dot\,\frac{89}{233}\)	\(137^{\circ}\,30^{\prime}\,38.6^{\prime\prime}\)	\(137.51073\)
Limiting value	\(137^{\circ}\,30^{\prime}\,27.9^{\prime\prime}\)	\(137.5078\)

Turing’s Table 1 demonstrates the connection of the Fibonacci numbers to the
observed difference in angular position of approximately 137.5°.

Thus, if we denote the Fibonacci numbers as \(f_1,\) \(f_2,\) \(f_3,\) etc., then

\[137.5078^{\circ}\approx 360^{\circ}\left({\lim_{n\rightarrow\infty}}\,{\frac{f_n}{f_{n+2}}}\right).\]

The following illustration drawn by Turing himself, labeled as the “daisy ring diagram,” illustrates a similar relationship between Fibonacci numbers and differences in position.  It is not explicitly linked to the document we consider here.

Turing's “daisy ring diagram” provides another example of the Fibonacci numbers in plant life.
(Source: Alan M. Turing Digital Archive, Archives Centre, King's College, Cambridge University,
UK.  Image AMT/K/3, page 3.  Copyright © P. N. Furbank.  Reproduced with permission.)

If we order the points based on their distance to the center, then the central angle between consecutive points is about 137.5°.  For more on Fibonacci phyllotaxis in general, and on this diagram in particular, see the articles by Swinton.

In the draft typescript, Turing discussed other Fibonacci-like patterns that occur in the parastichy numbers of different plant species.  The paper continues, covering material beyond the scope of this article, and so I will move on to discussing the student’s work and refer the interested reader to the original document itself.

Student Work: “Report” Group

In her paper, the student gave brief biographies of both Fibonacci and Turing and described the latter’s interest in phyllotaxis and its numerical intricacies. In addition to explaining much of the content, she also used diagrams to explain cylindrical coordinates and the notion of parastichy.  Moreover, she gave a short history of the study of phyllotaxis, including how its mathematical modeling was rejected outright by some botanists, all the way from the late 1800’s up until the time when Turing was writing this paper.  She even mentioned the continuation of this research after Turing’s death, citing work of Roger Jean and the development of the so-called “fundamental theorem of phyllotaxis.”  It was clear that the student thoroughly enjoyed this assignment.

While the student wrote a very engaging paper overall, there were certainly some areas where her paper could have been improved.  First, though interesting, the document she used was not written by Turing himself.  Instead, she could have used Turing’s own unpublished notes, which are quite similar to the edited document.  She also neither extended nor applied the mathematics beyond its initial scope.  As an example, she could have calculated some other limits involving Fibonacci numbers, such as finding \[360^{\circ}\left({\lim_{n\rightarrow\infty}}\,{\frac{f_n}{f_{n+k}}}\right)\]

for different values of \(k,\) and possibly found examples of some of these angles occurring in nature.  Or, she could have compared Turing’s original typescript to the typescript of Hoskin and Richards, to see what the latter pair had changed.  So while she found and interacted with a historical document of personal interest that contained Turing’s ideas, she did not attain the level of engagement that I was aiming to foster via the assignment.  Other students in the “report” group also wrote interesting, novel papers that were fun to read and well written, but came up short in the area of engagement.  On the contrary, the next example of student work, involving a paper of Euler, exemplified the kind of engagement I wanted the assignment to elicit.

Euler and a Problem from Number Theory

Title of Euler's paper in its original Latin, as it appeared in the Mémoires d l'Académie
impériale des sciences de St. Pétersbourg (Memoirs of the St. Petersburg Imperial
Academy of Sciences) in 1824.  (Source:  Digital image contributed to Biodiversity
Heritage Library by Natural History Museum Library, London)

Euler's E763 refers to and extends an earlier problem of Pierre de Fermat, namely, to find the two legs of a right triangle whose hypotenuse is a square and whose legs add up to a square.  Curiously, Fermat came across this problem while reading Claude Bachet’s 1621 translation from Greek to Latin of the Arithmetica of Diophantus (c. 250 CE), who posed the original problem. Mathematically, then, the original problem was to find two numbers whose sum is a square and whose squares add up to a fourth power.  First, Euler mentioned that Fermat and Lagrange had studied the problem, and then described his own early difficulties with trying to resolve the question of finding three integers with the same properties.  This led him back to Fermat’s original problem, the smallest solution of which involves integers greater than a billion.  Despite this initial setback, Euler was ultimately able to invent a general approach to finding solutions to problems of this type.

Euler’s Technique

In a nutshell, Euler exploited the Euclidean formulas for generating Pythagorean triples.  He first stated the original problem: to find \(x\) and \(y\) so that \(x + y\) is a square and \(x^2 + y^2\) a fourth power.  In his own words (using \(xx\) for \(x^2\), etc.) [§.5.]:

Let us begin with the latter condition.  At first, indeed, the formula \(xx + yy\) shall be rendered as a square, by placing \(x = aa-bb\) and \(y = 2ab,\) for then \(xx + yy = {(aa + bb)}^2.\)  In addition, then, this formula \(aa + bb\) should be a square, which happens in the same way by setting \(a = pp-qq\) and \(b = 2pq\):  from here, it follows that \(xx + yy = {(pp + qq)}^4,\) and thus the latter condition has now been fully satisfied.

Thus Euler used nested formulas for generating Pythagorean triples to guarantee that, regardless of his choice of \(p\) and \(q,\) \(x^2 + y^2\) would be a fourth power.  So then, he needed \(x + y\) to be a square.  After a bit of effort, he determined that \(p = 1469\) and \(q = 84,\) leading him to a solution [§.7.]:

These numbers are therefore

\(x =  4,565,486,027,761\)

\(y =  1,061,652,293,520\)

which are the same that Fermat, and others after him, found.  The sum of them is the square of the number \(2,372,159,\) while the sum of the squares is the fourth power of the number \(2,165,017.\)

For the three-variable version of the problem, Euler adapted the Euclidean formulas [§.8.]:

Let us begin again from the sum of squares, which is first rendered as a square, by placing \(x = aa + bb-cc; \) \(y = 2ac; \) \(z = 2bc; \) which thus will become \(xx + yy + zz = {(aa + bb + cc)}^2; \) whereby thus \(aa + bb + cc\) ought to be made a square again, which will be done in a similar way by putting \(a = pp + qq-rr;\) \(b = 2pr;\) \(c = 2qr;\) for thus is obtained \(xx + yy + zz = {(pp + qq + rr)}^4;\) so that the latter condition is now fulfilled.

Next, Euler found a relationship between \(p,\) \(q,\) and \(r,\) namely that \(p=r+{\frac{3}{2}}q,\) for which the corresponding values of \(x,\) \(y,\) and \(z\) would add up to a square.  His first example was \(x = 409,\) \(y = 152,\) and \(z = 64\) (corresponding to \(q = 2,\) \(r = 1,\) and thus \(p = 4\)), the sum of which is \(625 = 25^2\) and the sum of whose squares is \(194,481 = 441^2= 21^4.\) 

Euler handled the case of finding four numbers in much the same way, and found a similar relationship among his lowest-level variables, namely: \(p=s+{\frac{3}{2}}r-q,\) which would guarantee a solution.  For the case of five variables, Euler found that  \(p=t+{\frac{3}{2}}s-r-q\) would lead to a solution.  He then stated [§.23.]:

and thus for the case of six numbers it will be found that, \(p=u+{\frac{3}{2}}t-s-r-q,\) and so forth, from which the general question, proposed for any number of numbers, must now be considered completely solved.

As was the case with Turing, Euler went even further, but since his continued explorations are beyond the scope of this article, I will move on to the student work.  Interested readers may consult the original paper. 

Student Work: “Engaged” Group

In her paper, the student explained what Euler did, step by step from the beginning, and she even filled in some of the missing computational details behind his explanation of the three-number (\(x\)-\(y\)-\(z)\) case.  She then pointed out his general pattern and proceeded to use it to find six numbers with the same properties.  As her lowest-level variables, she selected \(u = t = r = 2,\) and \(s = q = 1,\) which meant \(p = 1.\)  This led her to the following six numbers:  \(97,\) \(112,\) \(64,\) \(64,\) \(128,\) and \(64,\) whose sum is \(529 = 23^2,\) and whose squares add up to \(50,625 = 225^2 = 15^4.\)

This student did what I expected.  She found a primary document, explained its content, and extended the mathematics beyond its original scope.  However, when selecting a topic, this student had struggled.  After a few unsuccessful attempts at finding primary source material that interested her personally, I eventually steered her toward E763, largely because I had just finished a draft of its translation from Latin to English.  The choice was convenient for her and for me, but I doubt that it piqued her curiosity.  Rather, she was doing what was asked, even if she was perhaps personally uninterested in the material.  Because this student’s experience was so different from the student who wrote about Turing and morphogenesis, I decided to look in more detail at the assignment itself.

Observations

One of the advantages to this kind of an open-ended assignment is that it can be tailored to individual students at their level of mathematical ability.  The two students whose work is featured in this article are probably in the average to good range, and so I wanted them to work with mathematics that was more within reach than, say, Leibniz’ proof of the Fundamental Theorem of Calculus. Turing’s mathematics was accessible and Euler used only algebraic techniques in this paper, and so both students made good choices in this regard.  Each student explained the mathematical content of her selected source material well. 

Some might argue that the mathematical content in these two selections was not “challenging” enough for an upper-division math course, and at first I would have agreed with them.  But while reading mathematical papers filled with algebraic equations has become second nature for us professional mathematicians, it is far from being so for undergraduates, even for math majors.  Thus, even if the mathematics in these papers was minimal or “easy,” it was still a challenge for each student to try to understand and fill in the missing steps, and I feel that both of them also did well in this aspect.

To contrast the students’ performances, however, the first student found a document that interested her, but she did not extend or apply the mathematics beyond its initial scope.  Rather, her paper read like a report on phyllotaxis and Turing, albeit a fascinating one.  On the other hand, the second student had difficulty finding a document that interested her, but once she had a topic, she did stretch herself by applying the formulas in that paper to create her own unique example, thereby engaging with the content in a way that I wanted students to engage.  To me, this assignment should embody the best of both worlds, by encouraging students to seek out primary sources of personal interest to them and then to converse with the ideas presented therein to put their own intellectual stamp on someone else’s work. 

Assignments like this can help to make mathematics and its history more meaningful to students as well as to faculty untrained in the history of mathematics.  Indeed, some of the shortcomings in the students’ writing can be attributed to my own inexperience with giving such assignments.  There is much to be gleaned from this experience. 

Planned Improvements

• Task Clarification.  To make sure each student poses and answers a mathematical history question, I need to be more explicit about what I mean by “engaging” with a primary source, ideally by sharing some model projects.  To be clearer, I want students to understand and explain the mathematics in their primary source, and then extend or apply that mathematics to something outside their primary source.  Next time, I will point students to what I feel are some particularly good examples of engagement, such as:  “Paradigms and Mathematics: A Creative Perspective” by Matthew Shives, a student of Betty Mayfield at Hood College, and also the winner of the HOM SIGMAA 2013 Student Paper Contest; or “The Moore Method: Its Impact on Four Female PhD Students” by Jackie Selevan, a student of Sloan Despeaux at Western Carolina University.  Several other good examples can be found in Despeaux’s article, "SMURCHOM: Providing Opportunities for Undergraduate Research in the History of Mathematics,” for Convergence.  There are also two outstanding examples of joint faculty-student projects directed by Adam Parker of Wittenberg University, “Peano on Wronskians: A Translation” and “An Analysis of the First Proofs of the Heine-Borel Theorem,” which can be found in Convergence here and here, respectively. 
• Document Selection.  To get the students to choose the primary sources themselves, I plan to start earlier.  And to help the students select appropriate source material, I will require them to fill out a brief checklist of basic properties of the document, such as: author, publisher (if published), date, location, provenance, etc.  This will enable the student (and me) to catch those sources that may not meet the strictest definition of “primary.”  On a related note, I plan to focus the entire course more narrowly on primary sources, and so that should provide the students with a wider variety of “acceptable” examples during the semester which they can then examine in more detail for their research papers.
• Early Draft. In addition to having students peruse the model examples listed above, I also will require a draft to be turned in several weeks before the final version, so I can make sure that they are trying to extend or apply the mathematics beyond its original scope.  I can also check the quality of their secondary sources, as students do not always have the information literacy required to distinguish peer-reviewed sources from those that are not peer-reviewed.
• Rubric Development.  As my goals for the assignment become clearer, this would also be a good time for me to develop a rubric that outlines what I am looking for in a way that is clear to the students and that will make the grading of the research papers objective, consistent, and understandable.  It could also help toward ongoing assessment efforts in the larger context of the math major.

Admittedly, many of these changes are fundamental features of a good writing assignment, and probably come second nature to those more experienced than I.  But perhaps, like my students, I needed to make a few mistakes of my own in order to learn my lesson.  By adopting these improvements in the future, I hope to push my students not only to select and report on a primary source, but also to engage more deeply with it and to enjoy a more positive learning experience overall.

References

Andre, Nicole R.; Engdahl, Susannah M.; and Parker, Adam E., "An Analysis of the First Proofs of the Heine-Borel Theorem." Loci: Convergence (August 2013), DOI:10.4169/loci003890 http://www.maa.org/publications/periodicals/convergence/an-analysis-of-the-first-proofs-of-the-heine-borel-theorem

Cooper, S. Barry and Van Leeuwen, Jan, eds.  Alan Turing: His Work and Impact.  Elsevier, 2013.

Despeaux, Sloan Evans, "SMURCHOM: Providing Opportunities for Undergraduate Research in the History of Mathematics." Loci: Convergence (January 2011), DOI:10.4169/loci003549 http://www.maa.org/publications/periodicals/convergence/smurchom-providing-opportunities-for-undergraduate-research-in-the-history-of-mathematics

Engdahl, Susannah M. and Parker, Adam E., "Peano on Wronskians: A Translation." Loci: Convergence (April 2011), DOI:10.4169/loci003642 http://www.maa.org/publications/periodicals/convergence/peano-on-wronskians-a-translation

The Euler Archive.  A digital library dedicated to the work and life of Leonhard Euler.  Accessed at http://eulerarchive.maa.org/ on September 26, 2013.

Euler, Leonhard.  "De tribus pluribusve numeris inveniendis, quorum summa sit quadratum, quadratorum vero summa biquadratum."  Memoires de l'academie des sciences de St.-Petersbourg 9, 1824, pp. 3-13.  Eneström index number E763.  Available at the MAA Euler Archive.  Accessed online at http://eulerarchive.maa.org/pages/E763.html on September 26, 2013.  Also available at the Biodiversity Heritage Library, as contributed by the Natural History Museum Library, London.  Accessed online at http://www.biodiversitylibrary.org/item/43682#page/61/mode/1up on June 1, 2014.

"Euler Portraits."  MacTutor History of Mathematics Archive.  Accessed online at http://www-history.mcs.st-andrews.ac.uk/PictDisplay/Euler.html on September 26, 2013.

Hoskin, N. E. and Richards, B.  “A morphogen theory of phyllotaxis.”  Draft. 1954. Archives Centre, King’s College, Cambridge. Turing papers AMT/C/8.  Accessed online at http://www.turingarchive.org/browse.php/C/8 on September 26, 2013.

Jean, Roger V.  “Number-theoretic properties of two-dimensional lattices.”  Journal of Number Theory, vol. 29, no. 2, June 1988, 206-223.

Klyve, Dominic; Stemkoski, Lee; and Tou, Erik, "Teaching and Research with Original Sources from the Euler Archive." Loci: Convergence (August 2013), DOI:10.4169/loci003672 http://www.maa.org/publications/periodicals/convergence/teaching-and-research-with-original-sources-from-the-euler-archive

Robson, Eleanor.  “Words and Pictures: New Light on Plimpton 322.”  The American Mathematical Monthly, Vol. 109, No. 2 (February 2002), pp. 105-120.

Sandifer, Ed.  How Euler Did It: “Euler as a Teacher - Part 2.”  February 2010.  MAA Online.  Retrieved online at www.maa.org/sites/default/files/pdf/editorial/euler/Feb2010.pdf on September 26, 2013. Available as of 2014 from MAA Euler Archive at http://eulerarchive.maa.org/hedi/HEDI-2010-02.pdf.

Selevan, Jackie.  “The Moore Method: Its Impact on Four Female PhD Students.”  In Sloan Evans Despeaux, "SMURCHOM: Providing Opportunities for Undergraduate Research in the History of Mathematics." Loci: Convergence (January 2011), DOI:10.4169/loci003549 http://www.maa.org/publications/periodicals/convergence/smurchom-providing-opportunities-for-undergraduate-research-in-the-history-of-mathematics

Shaw, Danny. “Royal pardon for codebreaker Alan Turing.” BBC News (24 December 2013) http://www.bbc.com/news/technology-25495315

Shives, Matthew. “Paradigms and Mathematics: A Creative Perspective."  In Janet Beery and Kathleen Clark, "HOM SIGMAA 2013 Student Paper Contest Winner." Loci: Convergence (July 2013), DOI:10.4169/loci003988

Swinton, Jonathan.  “Turing, Morphogenesis, and Turing Phyllotaxis: Life in Pictures.”  In Alan Turing: His Work and Impact, eds. S. Barry Cooper and Jan Van Leeuwen.  Elsevier, 2013.

Swinton, Jonathan.  “Watching the Daisies Grow: Turing and Fibonacci Phyllotaxis.”  In Alan Turing: Life and Legacy of a Great Thinker, ed. Christof Teuscher.  Springer, 2004.

Turing, A. M.  “The Chemical Basis of Morphogenesis.”  Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences, Vol. 237, No. 641. (Aug. 14, 1952), pp. 37-72.

Turing, Alan.  “Morphogen.  Theory of Phyllotaxis.”  Turing 2. Archives Centre, King’s College, Cambridge.  Turing papers AMT/C/25. Accessed online at http://www.turingarchive.org/browse.php/C/25 on September 26, 2013.

Turing, Alan Mathison.  Morphogenesis.  Edited by P. T. Saunders.  Elsevier, 1992.

Turing, Alan.  “Daisy ring diagram.”  Turing 2. Archives Centre, King’s College, Cambridge.  Turing papers AMT/K/3. Accessed online at http://www.turingarchive.org/browse.php/K/3 on June 12, 2014.

The Turing Digital Archive.  Accessed online at http://www.turingarchive.org/ on September 26, 2013.

“Turing Portraits.”  MacTutor History of Mathematics Archive.  Accessed online at http://www-history.mcs.st-andrews.ac.uk/PictDisplay/Turing.html on September 26, 2013.

About the Author

Christopher Goff is associate professor of mathematics at the University of the Pacific in Stockton, California.  In addition to math history, his mathematical interests range from the representation theory of Hopf algebras to the intersection of mathematics and popular culture.  He is a member of the MAA and the AWM, and has organized the on-site reception for LGBT mathematicians at the Joint Meetings for each of the last few years.  He is currently serving as the Chair of the MAA’s Golden Section.

References

Andre, Nicole R.; Engdahl, Susannah M.; and Parker, Adam E., "An Analysis of the First Proofs of the Heine-Borel Theorem," Loci: Convergence (August 2013), DOI:10.4169/loci003890 http://www.maa.org/publications/periodicals/convergence/an-analysis-of-the-first-proofs-of-the-heine-borel-theorem

Despeaux, Sloan Evans, "SMURCHOM: Providing Opportunities for Undergraduate Research in the History of Mathematics," Loci: Convergence (January 2011), DOI:10.4169/loci003549 http://www.maa.org/publications/periodicals/convergence/smurchom-providing-opportunities-for-undergraduate-research-in-the-history-of-mathematics

Engdahl, Susannah M. and Parker, Adam E., "Peano on Wronskians: A Translation," Loci: Convergence (April 2011), DOI:10.4169/loci003642 http://www.maa.org/publications/periodicals/convergence/peano-on-wronskians-a-translation

The Euler Archive.  A digital library dedicated to the work and life of Leonhard Euler.  Accessed at http://eulerarchive.maa.org/ on September 26, 2013.

Euler, Leonhard.  "De tribus pluribusve numeris inveniendis, quorum summa sit quadratum, quadratorum vero summa biquadratum."  Memoires de l'academie des sciences de St.-Petersbourg 9, 1824, pp. 3-13.  Eneström index number E763.  Available at the MAA Euler Archive.  Accessed online at http://eulerarchive.maa.org/pages/E763.html on September 26, 2013.  Also available at the Biodiversity Heritage Library, as contributed by the Natural History Museum Library, London.  Accessed online at http://www.biodiversitylibrary.org/item/43682#page/61/mode/1up on June 1, 2014.

"Euler Portraits."  MacTutor History of Mathematics Archive.  Accessed online at http://www-history.mcs.st-andrews.ac.uk/PictDisplay/Euler.html on September 26, 2013.

Hoskin, N. E. and Richards, B.  “A morphogen theory of phyllotaxis.”  Draft. 1954. Archives Centre, King’s College, Cambridge. Turing papers AMT/C/8.  Accessed online at http://www.turingarchive.org/browse.php/C/8 on September 26, 2013.

Jean, Roger V.  “Number-theoretic properties of two-dimensional lattices.”  Journal of Number Theory, vol. 29, no. 2, June 1988, 206-223.

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About the Author

Christopher Goff is associate professor of mathematics at University of the Pacific in Stockton, California.