More Bolts with Nuts
By Ralph Abraham
Revised Friday, 13 August 1999

Some exemplary projects for inclusive math at the Ross School, grades 5-11. In this version we add "nutshells" or abstracts to the list of titles. These abstracts are rough drafts, hastily written from memory, without access to a library, are not in chronological order, and may be improved someday.

Contents

Grade 5. 
5.1. Stonehenge and the carved balls of Scotland
5.2. The golden ratio in the art  and architecture of ancient Egypt
5.3. The monochord and the scales of Sumer, musical arithmetic
5.4. The rope-stretchers of Egypt and the orientation of temples
5.5. The squareness of the pyramids
5.6. The relation of Phi and Pi
	5.7. Musical arithmetic of the Rg Veda
	5.8. Binary arithmetic of the I Ching

Grade 6.
	6.1. The monochord and the Greek modes
	6.2. The golden ratio in Greek art and architecture
	6.3. From Pythagoras to Euclid: the root of 2
	6.4. The value of Pi
	6.5. Dimensions of the Temple of Solomon
	6.6. The ektar (or monochord) of ancient India and the 72 Carnatic melas (modes)
	6.7. Arithmetic in Phoenician notation

Grade 7.
	7.1. The machines of Archimedes, Heron, and Ctesibius
	7.2. Hypatia and the conics of Apollonius
	7.3. Tables of chords and astronomy in India
	7.4. The swath of trig from Ptolemy to Baghdad
	7.5. The logic of Aristotle's Isagoge
	7.6. Astronomical alignments in Teotihuacan
	7.7. The heliacal rising of Venus and the Mayan calendar
	7.8. Neoplatonism and Euclid in the Platonic Academy of Byzantium
	7.9. Early translations of Euclid in Baghdad, shades of Babylon

Grade 8.
	8.1. The Indian numerals 
	8.2. The constructions of Euclid 
	8.3. Sacred calligraphy and the birth of Al-jabr
	8.4. The 3-sphere in Dante's Divine Comedy
	8.5. Algebra of Ibn Tusi
	8.6. African divination
	8.7. Proportions of Gothic cathedrals, ancient geometry
	8.8. Geometry of Giotto
	8.9. The Hermetic Corpus in Ficino's academy, astrology and optics

Grade 9.
	9.1. Perspectiva from Giotto to Cosimo
	9.2. Conics from Apollonius to Durer
	9.3. Galileo and the pendulum clock
	9.4. Vincenzo and equal temperament, the 12-th root of 2
	9.5. Marin Mersenne and musical arithmetic
	9.6. Copernicus, Kepler, Newton: will Halley's comet destroy us?
	9.7. Cartography, Mercator, etc
	9.8. John Dee, Euclid, Shakespeare, the Sidney circle, the Royal Society
	9.9. Double entry bookeeping, John Recorde, logarithms
	9.10. Cartesian coordinates and the notations of algebra
	9.12. Kepler's Mysterium Cosmographicum and the Platonic solids
	9.13. Leibniz, historiography, binary arithmetic, the philosophers' war

Grade 10.
	10.1. The glass harmonium, Ben Franklin, Mozart, Chladni, Sophie Germain
	10.2. The heat equation, Fourier, Sophie Germain
	10.3. The wave equation, the hanging chain, more musical arithmetic
	10.4. Euclid's fifth postulate, measuring the three angles
	10.5. The shape of the earth
	10.6. Algebra in India

Grade 11.
	11.1. Cantor, his numbers, his fractals
	11.2. Hilbert's program, Godel's discovery
	11.3. More noneuclidean geometry and logic
	11.4. Poincare's Oscar, the rebirth of chaos, Sofia Kovaleskaya
	11.5. Fractals of Julia, paintings of Kupka
	11.6. The foundations of calculus, Dedekind and the real numbers
	11.7. Flatland, differential geometry of curves and surfaces
	11.8. German vs French math styles, Hilbert's program, Godel
	11.9. Carnap and the Vienna Circle
	11.10. Cantor, infinity, the first fractal
	11.11. Hertz, radio waves, the field concept, vectors, Maxwell equations
	11,12. Hamilton, quaternions, Minkowski, Poincare, Einstein, the ether
	11.13. Vibrating strings and quantum mechanics, particles vs waves
	11.14. Lewis Frye Richardson, the birth of politicometrics, Gregory Bateson
	11.15. Ludwik Fleck, paradigm shifts
11.16. Samuel Rashevsky
	11.17. Von Neuman, game theory, the atomic bomb, prisoners dilemma

Nutshells of exemplary units for Grade 5. 

5.1. Stonehenge and the carved balls of Scotland

The megalithic  culture built the ancient stone monuments of Brittany and the British Isles. This story is contemporary  with the Riverine civilizations, was probably influenced by Mesopotamian  and/or Egyptian scholars, but was prehistoric. The Avebury Circle and Stonehenge, perhaps the best known megalithic  structures, contain solar and celestial allignments. The henge at Callanish on the Scotish island of Lewis has been shown to be a lunar observatory. Among the least known artifacts of this amazing culture are the bethels, or carved stone balls, of Scotland. Some hundreds of these handiworks, about the size of a tennis ball, have been found all over Scotland, and are collected in the Museum of Antiquities in Edinborough, where they may be seen today. They are carved from a rather hard stone by expert stone masons, and our interest in them is due to their shapes. All present rather sophistaicated knowledge of solid geometry, and no doubt they are part of a mathematical tradition which culminates in Pythagoras and Euclid.

The idea of this unit is to appreciate the possibility of prehistoric math and archeoastronomy contemporary with the pyramids and ziggurats, so speculate on the direction of diffusion North or South, and in any case to appreciate the content of Euclid's Elements and its pre-Greek evolution. 

In a nutshell: prehistoric (megalithic) geometry and archeoastronomy contemporary with the Riverine civilizations.

5.2. The golden ratio in the art and architecture of ancient Egypt

The golden ratio, Phi, is an irrational number, approximately 1.618. The golden rectangle , which has sides in the proportion of Phi, is strangely attractive the human aesthetic sense, according to numerous modern psychological experiments.  Apparently, the ancients knew this, and they evolved geometrical methods to construct golden rectangles. One of these constructions is found later in ancient Greece, and figures importantly in Euclid's Elements. It is amazing to find this construction in a bas relief of ancient Egypt, and thus, it is appropriate to study this construction in Grade 5, and to examine artifacts of the Eriverine civilizations searching for Phi, as we already do in Greade 6.

In a nutshell: plane geometry may be discovered in Riverine art and architecture.

5.3. The monochord and the scales of Sumer, musical arithmetic

The monochord is a one-stringed musical instrument, or more properly, a laboratory instument for musical arithmetic. Studying musical intervals, harmony, scales and modes on the monochord is a thread weaving through all of the grades. In Grade 5, in the ambiance of Babylon or Memphis for example, one might study one song, one musical instrument, one scale, and introduce the octave, and the intervals of perfect fifth and fourth, and the arithmetic: (4/3) * (1/4) = 1/3.
5.4. The rope-stretchers of Egypt and the orientation of temples
Just one of the reasons that the Great Pyramid was considered one of the seven wonders of the world is the accuracy of its orientation to true North. This feat required a combination of skills from astronomy and from geometry. The recreation of this multidisciplinary  provides a unique experience of the Riverine cultural ecology, furthers the thread of archeoastronomy, and begins the thread of Euclidean geometry.
5.5. The squareness of the pyramids
Another astonishment of the Great Pyramid is the accuracy of its dimensions. Another ancient construction which found its way into Euclid's Elements, and a connection perhaps with the value of Pi.
5.6. The relation of Phi and Pi
This is a long shot, perhaps in the "pyramidiot" category, due to Schwaller de Lubitz. According to this theory, the ancient Egyptians knew the values of both Pi and Phi sufficiently to establish a close approximation to their ratio. This unit furthers the concept of irrational numbers, and an appreciation of the sophistication of the master builders of antiquity.
5.7. Musical arithmetic of the Rg Veda
This project depends on the work of McClain, and builds upon the Sumerian monochord project, 5.3 above, with which it may be compared. The evidence for early musical arithmetic is encoded as unusual, large integers, and provide drill in mental arithmetic. For the precedence of arithmetic to language, see Deheane.
References. 
McClain, Ernest, The Myth of Invariance.
Deheane, Stanislas, The Number Sense. 
5.8. Binary arithmetic of the I Ching
The I Ching has evolved from divination practices of the Shang, and consists of hexagrams with divinitory annotations. The hexagrams appear to be numbers written in binary notation using stick symbols derived from the counting board. Chinese counting boards and stick figures are dexcribed in Meninger, Number Signs and Number Symbols. Leibniz was fascinated by the I Ching, and developed binary arithmetic from it.
Grade 6.
6.1. The monochord and the Greek modes
In this grade we wish to develop the ideas of the quadrivium of arithmetic, geometry, astronomy, and music. The music of the quadrivium was what we now call musical arithmetic, and has been reconstructed by classicists and musical scholars such as Robert Brumbaugh and Ernest McClain.  Some of the ideas of tuning theory -- intervals, scales, keys, modes, lemmas, commas, etc -- would be explored on the monochord, depending on the skills and interests of the teachers and students: this is a difficult subject. Hopefully one would progress at least to the understanding of just intonation and the ten modes of ancient Greece. This unit begins with the biography, ideas, and influence of Pythagoras, and furthers a thread begun in Grade 5.
6.2. The golden ratio in Greek art and architecture
What is the golden ratio and why do we care? While hardly anyone today can answer these questions, every educated person in classical Athens could. We prove this by measuring a number of Greek art works, vases, and architectural monuments, for example, the Parthenon.
6.3. From Pythagoras to Euclid: the root of 2
This is a tough nut to crack: Why were the Pythagoreans so bamboozled by the discovery that root 2 is not a fraction? This connects with the monochord thread and to geometric algebra, and represents a major bifurcation in the evolution of consciousness. We try to appreciate the paradigms of the ancient world both before and after this discovery.
6.4. The value of Pi
Archimedes, in the time frame of Grade 7, had a most clever idea here, which played a role in the discovery of the calculus. This unit advances the thread begun in Grade 5, connects it to Grade 7, and develops the appreciation of Euclid's Elements.
6.5. Dimensions of the Temple of Solomon
The dimensions of the Temple of Solomon, ca 950 BCE, entranced Newton. This is more practice in the arithmetic skills, and the appreciation of the arithmetic mentality.
6.6. The ektar (or monochord) of ancient India and the 72 Carnatic melas (modes)
More musical arithmetic, rather easier than McLain's large numbers. One may appreciate the development of the arithmetic mentality  as derived from musical evolution.
6.7. Arithmetic in Phoenician notation
Here we touch upon the history of writing and media, with the invention of the first alphabet, and its subsequent application to the project of recording arithemetic, or vice versa.
Grade 7.
7.1. The machines of Archimedes, Heron, and Ctesibius
Lots of creative engineering in antiquity, such as the simulacron which was the alarm clock of Plato's academy, the cannons of Archimedes, and so on. Technology should be touched upon in each grade. In this unit, we learn of the steam engine and pipe organ of Heron of Alexandria, and connect with the monochord thread, musical arithmetic, and the history of musical instruments.
7.2. Hypatia and the conics of Apollonius
The conics of Apollonius will surface again in Grade 9, in the perspective studies of artists and architects, and in Grade 10, where they become approachable by meer mortals thanks to the coordinate system of Descartes. In Grade 7, we may take this opportunity to study the integration of math and philosophy in the Museion or Library of Alexandria, one of the first universities, the roles of Theon, Hypatia, and Proclus in packaging the Greek corpus for its journey into the Middle Ages, and the disintegration of all this after the murder of Hypatia by a Christian mob, and the burning of the Museion by the Muslim conquerers.
7.3. Tables of chords and astronomy in India
Following Ptolemy to India and beyond, the development of trigonometry, astronomical tables, periods of the sun and moon in calendrical arithmetic, other aspects of Indian culture.
7.4. The swath of trig from Ptolemy to Baghdad
Migration of Hindu math and astronomy, the Silk Road, Merv, Persian culture. Could be combined with 7.3 above.
7.5. The logic of Aristotle's Isagoge
We assert that a certain balance of the rational and the irrational in ancient Greek philosophy, represented by Aristotle and Plato respectively, or rather the Aristotelians and the Neoplatonists, was lost in the transmission of the Greek corpus to Europe, due to the special interest of the Muslims in Aristotelian logic. With the Isagoge, we get a first instroduction to formal logic from the original source.
7.6. Astronomical alignments in Teotihuacan
Teotihuacan boggles the mind, as a totally astrogeometrical city plan. The dimensions of the pyramids and many other features hae been studied. In this project we consider only the simplest features of the Teotihuacan plan: the astronomical allignment of its major axis.
References.
Tompkins, Peter, The Pyramids of Mexico.
7.7. The heliacal rising of Venus and the Mayan calendar
The Mayan calendar was more accurate than our own. This project develops respect for the high cultures of the Americas, their base 20 counting system, arithemetic mentality, writing systems, calendars, and astronomy. The heliacal rising of Venus may be seen directly in the winter mornings at the school.
7.8. Neoplatonism and Euclid in the Platonic Academy of Byzantium
The Neoplatonists may be reviewed very concisely, with emphasis on Plotinus, Iamblichus, Hypatia, and Proclus. The closure of the Platonic Academy in Athens, brief sojourn of the mathematicians in Persia, founding of Constantinople, and the openning of the new Platonic Academy there. The biography of Leo the mathematician may be compared with the story of Democratus in Alexandria, and the removal of Aristotle's library.
7.9. Early translations of Euclid in Baghdad, shades of Babylon
Besides the stories of the translations of Euclid into Syriac and Arabic, we may speculate on the vestiges of Babylonian arithmetic and geometric algebra in early Islam. This compares two layers of a palimpsest, and introduces some interesting ideas of clutural diffusion.
Grade 8.
8.1. The Indian numerals
This is a good time  to begin the history of alphabets. In this context, the Syriac alphabet reached India, turned into the Brahmi script, and spawned the Indian number symbols, which reached Baghdad around 810 CE, as we know from the first volume of Al-Khowarismi. This journey fits in the context of Grade 7, but the further journey of the numerals to Europe follows the expansion of Islam across north Africa , where they were discovered by Leonardo of Pisa (Fibonacci) and brought to Palermo and then Rome arund 1200 CE. This story should be touched either at the end of Grade 7 or early in Grade 8. Along with the story, pupils would do arithmetic drills in the notations of ancient Greece, Canaan, and Rome (Roman numerals) before 1200.
8.2. The constructions of Euclid
This is the story of geometric algebra, and the birth of Arabic algebra, and a nontoxic introduction to the rhetorical algebra of the Middle Ages, modernised later (1637) by Descartes. This thread will catch up with modern algebra in Grade 9, hopefully circumventing the math anxiety frequently created by the introduction of symbolic notations. In this sequence, an area problem from Babylonia will be discussed in its original Babylonian style, then again as a construction of Book I of Euclid's Elements, and finally in the style of Al-Khowarizmi. Victor's book and my chapter on Al-Khowarizmi may be consulted. This unit may be very helpful in regard to math anxiety, multiple intelligence  styles, and gender differences. There are some materials online relating to this unit.
8.3. Sacred calligraphy and the birth of Al-jabr
The preceeding unit integrates the conventional writing arts, Eucliean geometry, and algebra. This unit is very similar, but in addition brings in calligraphy as a spiritual practice, as it developed in China, Judaism, and Islam. Thus, Arabic language and Islamic culture are explicitly joined to Unit 8.2. These two units are supported by an extensive document with references: www.ralph-abraham.org/articles/MS#xx.
8.4. The 3-sphere in Dante's Divine Comedy
Ok then, in deference to the skepticism of Bruce, let us consider only the 2-sphere in Dante. Dante, circa 1310 CE, deescribes a cosmological model which may be interpreted as the construction of the 2-sphere (the unit sphere in Euclidean 3-space) as the join of two 2-discs. Pupils may read Dante's cosmology, and then construct a soccer ball from two flexible discs: heaven and hell. This model might be used to deconstruct any binary, such as good/bad, literate/illiterate, etc. Also, one has a chance to appreciate the genius of Dante and his role in the precipitation of High Gothic (eg, Giotto) and the Early Rennaisance (eg, Rabelais). Thus, geometry, literature, and painting are yoked, in preparation for the Perspectiva unit of Grade 9. This unit is also supported by a document available online.
8.5. Algebra of Ibn Tusi
In later Islam, the math and sciences of Baghdad diffuse into Mongolia. Here is an example of the diffusion and evolution of algebra, on the way to the subjects we now teach in conventional high schools.
8.6. African divination
Divination practices in the Isa (??) show the development of arithmetic in prehistorical cultures. This project has been taught rpeviously in Grade 8 by Gottfried Mayer-Kress, and some records may be found in the archives.
8.7. Proportions of Gothic cathedrals, ancient geometry
Endless material for projects here, consult any of the many texts for ideas regarding proportions, rose windows, labyrinths, and so on. The magisterial two-volume work by Brunes emphasizes the square root of two, called the silver section, rather than the better known golden section, related to the square root of five. How did the master builders actually construct the silver section? This question brings us once again the the constructions of Euclid.
8.8. Geometry of Giotto
Here we find the intuitive rediscovery of the perspective of ancient Greece, and the beginning of the breakthrough from the plane to three-dimensional space, laying the foundation for the Renaissance perspective unit of Grade 9.
8.9. The Hermetic Corpus in Ficino's academy, astrology and optics
Dame Francis A. Yates dug out the Hermetic roots of the Renaissance philosophers. In preparation for Grade 9, we may introduce the Hermetic Corpus here, as an Alexandrian and pseudo-Egyptian (or perhaps really Egyptian) literature. The content is a window into the occult shadow of all cultures: astrology, alchemy, magic, healing, and the like.
Grade 9.
9.1. Perspectiva from Giotto to Cosimo
Following the inspiration of classical Greek materials, the application of geometry to drawing, painting, and architecture was developed by artists from 1300 to about 1450 CE. This is the best known example of "bolts from the blue", or new math, science, and technology derived from inspiration by artists, then passed on to science. A short document exists somewhere in the school archives, and this unit was taught successfully last year.
9.2. Conics from Apollonius to Durer
This is a further development of the rediscovery of classical Greek knowledge in the Renaissance, and a short document exists somewhere.
9.3. Galileo and the pendulum clock
The history of technology is an important thread which is perhaps underrepresented in our program. The clock is a foremost example of historically important technology. The water clock of ancient Egypt, the Athens town clock and planetarium, the alarm clock of Plato's academy: all are fascinating precursors to the discovery af the pendulum clocks, the first which kept time reliably. The idea if the escapement mechanism is due to Galileo, and the first working exemplar to Huyghens, and this story thus belongs to the time frame of the end of Grade 9 or the beginning of Grade10. Galileo discovered the isochronous property of the pendulum early in life and used the pendulum as a timer  in his famous experiments in dynamics. This unit should begin in Grade 9, and be connected to those experiments, and the founding of the science of dynamics by Galileo. The unit could be continued in Grade 10 when the concepts of the calculus are introduced.
9.4. Vincenzo and equal temperament, the 12-th root of 2
The thread on musical arithmetic and the unit on the irrationality of root 2, from earlier grades, come together here. For the relevant history, one may consult any of the numerous texts on the history of the musical scale. For the integration of mathematics, one might try to approximate the frequencies of the chromatic scale by calculation by hand, checking with a hand calculator, and finally, resorting to logarithms. (Is this in Grade 10?)
9.5. Marin Mersenne and musical arithmetic
The discovery of Pythagoras on the relation between the length of a string and its fundamental frequency of vibration when plucked or bowed was improved by Mersenne (and others??) through experiments involving the thickness of the string and its tension. This is opportunity to enter into the science of acoustics, or the physics of sound, at an early stage. Later, this thread will be continued with the work of Ernst Chladni, the father of acoustics.
9.6. Copernicus, Kepler, Newton: will Halley's comet destroy us?
This story is retold on all texts of the history of the calculus, the biographies of Newton, etc. The conics of Apollonius reappear as trajectories of a dynamical system, and the dynamical mentality is born. This is one of the primary bifurcations in the Thompson scheme of the four cultural ecologies of the West, which in our curriculum, demarks the boundary between Grade 9 and Grade 10. Without this unit, in fact, the whole scheme is occult.
9.7. Cartography, Mercator, etc
The applied mathematics of the Late Renaissance involved cartography, navigation, and other practical arts, as we know from Dee's preface to Billingsley's Euclid. Frisius and Mercator in Louvain represent a midpoint in the evolution of our idea of the Earth from the ancients (Eratosthenes and Ptolemy) to Copernicus. Exercizes in this project could involve joining local charts, in Photoshop for example, into a map of the world. Mercator and other projections from the 2-sphere to the plane would be understood from these experiments. Cartography, like the history of clocks or calendars, is a royal road to geometry, and the evolution of consciousness.
9.8. John Dee, Euclid, Shakespeare, the Sidney circle, the Royal Society
Hot time in Merry olde England. Dee, the outstanding mathematican of his time, Imperial Mathematicus to Queen Elizabeth I, founder of the Royal Society, spy, astrologer, conjurer of angels, architect of the British colonial empire, thief of America, creator of Enochian magic, alchemist… is perhaps the most interesting biography on record.
9.9. Double entry bookeeping, John Recorde, logarithms
It is through business applications that mathematics found its way into universities in England, and John Recorde is the point man for this bifurcation, which leads directly to Barrow, Newton, and the calculus.
9.10. Cartesian coordinates and the notations of algebra
Descartes made several important contributions to mathematics. Stimulated by the rediscovery of perspective in the Italian Renaissance perhaps, he introduced Cartesian coordinates (sounds like, and looks like, cartography, but refers to Descartes name instead) and changed geometry from a post-graduate research program into a middle school subject.
9.12. Kepler's Mysterium Cosmographicum and the Platonic solids
While Euclid's Elements begins with the mathematics in the Riverine civilizations in chronological order, it ends with the five Platonic solids. One should know these, feel them, and understand why there are only five. I cannot give a reason for this, it is simply a Must Know. These figures were known in prehistoric (megalithic) cultures, as described in a project for Grade 5. Beginning with the mind frame of these objects, Kepler imagined God's work in the creation of our planetary system as a Euclidean construction, as described in his first book, the Mysterium Cosmographicum. Trying to prove his theory led him to his lifework, and his three laws, another Must Know.
9.13. Leibniz, historiography, binary arithmetic, the philosophers' war
Grade 10.
10.1. The glass harmonium, Ben Franklin, Mozart, Chladni, Sophie Germain
Domains: PA (music), Science (acoustics), math (calculus)
Threads: Musical arithmetic, American history, vibrations, gender issues
Time frame and place: France, Germany, ca 1800
Nutshell:
A new musical instrument. Ben went to Paris and consumed quite a bit of wine. While the habit of twiddling the goblet in the left hand while rubbing the rim with one finger of the right was practiced by all experienced winos, Ben developed the habit into a serious musical instrument called the glass harmonium. In it, a whole series of wine goblets, deprived of their stems, whre mounted collinearly on a horizontal metal rod. Spun with a treadle like a sewing machine and suspended over a pan of water, the motion and wetness of the glass was automatic. One need only touch the moving wet rims with a finger, and play the thing like a keyboard instrument. Musicians were fascinated then, as now, with new technologies, and composers and players flourished. Even Mozart composed for a the glass harmonium.

The problem. The timbre of the glass harmonium was quite annoying, rather worse than the musical saw or the Theremin machine. Chladni sought to improve the timbre by replacing the goblets by circular disks of glass. To this end he experimented with disks of various thicknesses and sizes, using a cello bow applied to the edge of the disk to produce the sound. This is an pre-electonic analog of the timbre programing of our electronic synthesizers. He observed, as had Leonardo before him, the patterns of dust moving on the fixed, horizontal disk, during the bowing. These are related to the crispations of Faraday, discovered on a barrel of beer by Faraday, in a pub, a few years later.

The patterns. Sprinkling sand on the disk to improve the visibility of the patterns, he discovered the patterns of ridges we now call the Chladni nodal lines. He travelled around Europe giving live demostrations, was introduced to the French Academy by Laplace, and had a private audience with Napolean. Nappy was so impressed he announced a prize for a mathematical explanation of the nodal lines.

The prize. Sophie Germaine won the prize, and this is an opportunity to review the state of sexism in math and science during this period of history. 
References: 
Biography of Sophie Germaine, Ben Franklin, search web for glass harmonium (still made), history of acoustics, see Chladni in Gillispie.
10.2. The heat equation, Fourier, Sophie Germain
Domains: Science (physics), Math (calculus)
Threads: Gender issues
Time frame and place: France, Egypt, ca 1810

Nutshell:
Fourier is an interesting personality in the history of France. A member of Nappy's expedition to Egypt and active politician, he obtained his seat in the French Academie with some difficulty. His derivation of the heat equation earned him a place in the math Hall of Fame. After the premature death of Sophie Germaine, their relationship was revealed. One may speculate about the role of Germaine in Fourier's work.
References: Biographies of Fourier, Germaine.
10.3. The wave equation, the hanging chain, more musical arithmetic
Soon after the publication of Newton's work, dynamical systems theory was advanced step-by-step from the simple pendulum to the double pendulum, the hanging chain, the dangling rope, and finally the stretched string. The resulting model, the wave equation, has been the mainstay of mathematical physics ever since. Its author, d'Alembert, escaped the guillotine, and was a leading contributor to the Encyclopedie project of the Enlightenment. Versions of the wave equation, especially the generalization of Sophie Germaine, gave a satisfactory understanding of Mersenne's equation for the sound of a stretched string.
10.4. Euclid's fifth postulate, measuring the three angles
Questions about the neccessity of the fifth postulate of Euclid were raised soon after the time of Euclid, and persisted throughout the Middle Ages. Finally, Euler (ca 1750) secretly wondered if God had created the world flat or not. He set out to answer this question experimentally, and neasured the angles between the lines of sight connecting the three tallest church steeples of Basel. I am suggesting here the reproduction of this experiment.
10.5. The shape of the earth
This story, aka the battle of the bulge, is told concisely in my Dynamics, the Geometry of Behavior, and in more detail in the references found therein. In a nutshell, England and France nearly went to war over the question: is the Earth lemon-shaped, or melon-shaped? This was resolved by two expeditions, one to the Equator, the other toward the North Pole. The invention of sufficently adequate clocks made these expeditions possible.
10.6. Algebra in India
Besides an opportunity to rview the unique mathematical heritage of India, this provides an opportunity to revisit the threads of musical mathematics, the arithmetic mentality, the stimulus of Euclid and Ptolemy, the tramsmission from Arabs such as Al-
Tusi, and the effects of the English colonization. See G. G. Joseph for all of this.
Grade 11.
11.1. Cantor, his numbers, his fractals
The transfinite numbers of Cantor, definition of infinity, set theory, the middle thirds set, his biography.
11.2. Hilbert's program, Godel's discovery
Hilbert's rewprking of Euclid's postulates, his formalist program and address of 1900, the liar's paradox, idea of Godel's proof.
11.3. More noneuclidean geometry and logic
The independence of the 5th postulate, Bolyai and Lobachevsky, Tom Lehrer's song, formal logic.
11.4. Poincare's Oscar, the rebirth of chaos, Sofia Kovaleskaya
Dirichlet, Weierstrass, Sofya Kovalevsky and Anna Carlotta Leffler, Poincare, King Oscar II, idea of the chaotic tangle and the death of laplacean determinism, chaos theory.
11.5. Fractals of Julia, paintings of Kupka
Anticipation of Julia's 1920 discovery of fractal geometry in the 1910 paintings of Frantisek Kupka.
11.6. The foundations of calculus, Dedekind and the real numbers
Mid nineteenth century Berlin, the culmination of the calculus, the completion of the foundations of arithmetic, the development of opposed mathematical styles in Germany and France, and the role of mathematics in the great bifurcation into the Modern.
11.7. Flatland, differential geometry of curves and surfaces
From the intuitive understanding of dimension from Flatland, we move into the new differential geometry, connect with noneuclidean geometry, and model these latrernative geometries on curved surfaces, and apply to a simple introduction to general relativity.
11.8. German vs French math styles, Hilbert's program, Godel
Going forward from the bifurcation of math styles into the 20th century, the beginning and end of the formalist program, the nature of mathematics, the beginning of mathematics in the Americas.
11.9. Carnap and Schlick in the the Vienna Circle
Except for Godel, the Vienna Circle was not about mathematics directly as much as about logic and the philosophy of science. But here we may see the beginning of the end of the partnership of math and science which had characterized our cultural ecology during the arithmetic, geometric, and dynamical mentalities, and prepares the way for the chaotic dynamical breakthrough.
11.10. Cantor, infinity, the first fractal
As the algebraic and geometric streams culminate in the partnership of set theoretic  and algebraic topology, bringing the modern phase of mathematics to a close, the postmodern makes a premature appearance in the perfect sets of Cantor. What is the significance of Cantor's middle thirds set, its implication for the mathematical modeling of nature, and the role of all this in Cantor's mental illness?
11.11. Hertz, radio waves, the field concept, vectors, Maxwell equations
The field concept emerged in the discovery of electricity (see Shlain for this) and took its mathematical form in the concept of vector field, with which Maxwell modeled the fields made visible by Hertz with his Tesla coil and flourescent tubes. From this came radio and television broadcasting, the tensor geometry basic to general relativity, the matrix algebra fundamental to quantum mechanics, the discovery of chaos by Poincare, etc etc etc.  Again, modern math culminates, and postmodern math begins, with fields of vectors and tensors.
11,12. Hamilton, quaternions, Minkowski, Poincare, Einstein, the ether
Another thread to follow through the bifurcation so key to this grade. From complex numbers to quaternions to the special relativity of Minkowski et al. Is there or is there not an ether? Did the Michaelson/Morley experiment settle the question for ever or not? Could we be living after all in a ten-dimensional universe? What is the good of having a mathematical model of the cosmos? Is cosmology actually cosmic? How can anyone really take the anthropic principle seriously? Do you think the universe is really expanding? Is the big bang theory the creation myth of our time? Is there any real good in studying math and physics on this level? Should the NSF support the space program? Do the unanswered questions of math and science stimulate cultural evolution? Should Beavis and Butthead really be discussing these things in front of children on television?
11.13. Vibrating strings and quantum mechanics, particles vs waves
Vibration is a metaphor from the ancient world, from bowstrings, ponds and streams, musical strings, sounds of waterfalls, bird songs, human speech and song, and so on. The vibrating string and the monochord were the springboards of ancient science, along with the appearances of the sun and moon, stars, and planets. AS the music of the spheres, the vibration metaphor reached for the stars, and the stars radiated good and bad influences down upon us. In this project we carry forward this thread from the beginnings of modern mathematical physics in the wave equation of d'Alembert, 1749, to the Einstein equation and Schrodinger equation, and try to make sense of the conundrum of wave-particle duality, solitons, etc.
11.14. Lewis Frye Richardson, the birth of politicometrics, Gregory Bateson
Richardson was a physicist involved with meteorology before the first war. His program for weather prediction was the most accurate of his time, and was used in England until recently. As a Quaker, he objected to military service, and was drafted as a consciensous objector in the first war. He served in the ambulance corps, and thus saw much grisle, and decided to devote the rest of his life to the prevention of war. His model for war created the field of erodynamics, aka system dynamics and social physics, as described in my book, Chaos, Gaia, Eross. The rejection of his work prompted his life long efforts to justify his model on the basis of actual data, which gave rise to the field of politicometrics. He lived to see the second war give further validation to his model, following which his works, such as The Statistics of Deadly Quarels,  were published posthumously. His model for war was adopted by Gregory Bateson as a model for schizophrenia, and research following his ideas continues today. (Eg, recent papers by Mayer-Kress and myself.) These models may be easily simulated using the STELLA program in our own tech room.
11.15. Ludwik Fleck, paradigm shifts, Samuel Rashevsky
Fleck was a child during the first war, became an important microbiologist, and was taken to Auschwitz during the second war. From the security of his laboratory window, he saw his family taken away to the gas chambers, while he continued his research on the prevention and cure of STDs such as syphilis. Between the wars he wrote his book, The Genesis and Development of a Scientific Fact,  which began the study of the sociology of science and introduced the important idea of paradigm shift, or catastrophic bifurcation, in the sciences. In this book he analyzed, as the paradigmatic case of paradigm shift, the Wasserman test for syphilis, that is, the interaction of bacteria and the immune system in disease etiology. He survived the war and returned to his home town near Bonn.
11.16. Samuel Rashevsky
Rashevsky made a daring escape, with his wife, from the Russian Revolution. Landing eventually in Chicago, he created the field of mathematical biology, began a graduate program which continues today as one of the best in this field, applied the ideas of that subject to world cultural history in hopes of promoting the cause of peace, and was the first to identify catastrophic bifurcation as a mathematical model for war.
11.17. Von Neuman, game theory, the atomic bomb, prisoners dilemma
Von Neuman, like Fleck, was a Jewish child during the first war. Unlike Fleck, he escaped the holocaust to safety in the US. Following his early work on the foundations of quantum mechanics (which I studied as a graduate student) he founded mathematical  economics.  He then became one of the main inventors of digital computers, as well as of the atomic bomb. Following the war he invented game theory. Toward the end of his life he wrote about robots that could reproduce themselves, and speculated on the similarity of computers and brains. He died prematurely just as my own career began. An application of game theory to the social sciences, the prisoner's dilemma, has spawned a whole industry, and may be applied in this unit to arms races, and to environmental problems.

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