Synopses of the for Teenagers set
The books have "for Teenagers" (or "for Preteens") in their titles because they are intended for high-schoolers anticipating university. They include material I have been teaching to 11--17-year-olds at a Montreal school. The intended audience is much broader and includes all interested people who have at least basic algebra (and not even that for the preteens book). Undergraduate students of physics, computer science and related fields may be a major part of that audience, overlapping to pre-undergrads, post-undergrads, teachers of STEM (science, technology, engineering and mathematics), and the interested general public as a whole.It is about physics (particle, collective, Newtonian) and computing, all of which are intrinsically mathematical. I am attempting to democratize the subjects, not popularize them. Popularization removes all the mathematics. Since the concepts are fundamentally mathematical, it is better to find a way to make the mathematics accessible.
To introduce all this in detail to people whose mathematics is limited to simple algebra - and that a little shakey - requires basic introductions to spatial math and angles, to sums and related areas, to histograms and related distributions and probabilities, to the mathematics of small changes, and of course to momentum and energy and their great conservation laws. (Not going beyond the algebra would permit unmotivated calculation but omit the mathematical ideas, on which the physical ideas are based.) The books start slowly but accelerate, so beginners reading later chapters may need support of an experienced mentor - but skimming the rest after some grounding in the early chapters will also be rewarding.
Each book is self-contained: they form a set, not a series.
Physics for Teenagers: Relativistic Quantum Physics
Synopsis
The quantum physics of polarized light requires a projection operator. The Lorentz transform of special relativity is a shear operator. Such operators are represented by matrices so we must start there and in particular with the unexpected math of matrix multiplication. But this is little more than arithmetic, and the 2-by-2 cases capture all the basic ideas.Particles can be periodic if represented by complex numbers. These are abstractions of rotations, another matrix form. So particles behave as waves: frequency gives energy and wavenumber gives momentum. The invariants of the Lorentz transformation then give rest mass and E=mc^2.
In more dimensions than two, rotations are better treated as pairs of reflections. These anticommute and abstract to Clifford numbers, which can be used to describe space in any number of dimensions. The idea of spin-1/2 follows from reflections being more fundamental than rotations. And ladder operators, a complex combination of reflections, can describe the creation and annihilation of particles at relativistic energies: quantum field theory.
The phase of the complex number describing a particle, which at first seems to have no physical consequences, accounts for the distinction between bosons and fermions, a distinction which applies to all particles. The circular symmetry of phase in complex numbers leads to gauge theory which accounts for the electromagnetic field. This extends to spherical symmetry, using three-dimensional rotation generators, which correspondingly extends electromagnetism to the electroweak force. A further extension of the symmetry leads to quarks and the strong nuclear force.
A final generalization of matrices gives relations and tensors and introduces the puzzle of entanglement and the parallelism of quantum computing.
The last chapter discusses proof in mathematics and how science can know things.
Audience: people who remember some algebra and are willing to learn about the mathematics of spaces of two and more dimensions, including angles, en route to the "Standard Model" of physics.
Under the hood
Spacetime: if I ask you how far you live from school and you say "20 minutes" or so, you've answered a space question with a time. So space and time mix somehow. The "somehow" assumes a velocity - e.g., "by bus" - so velocity mixes space and time. It gets complicated when we disallow infinite velocities. Then the inference that there is a maximum velocity turns out (after brief instruction on matrices) to imply that we can give space answers to time questions. Special relativity follows.Amplitudes: two polaroid sunglass lenses block varying amounts of light depending on their angle to each other; at right angles they block all light, but a third lens inserted between them lets some light get through the whole set. It turns out (after a slightly different brief instruction on matrices) that this can be understood by light having a state and a certain probability of being in another state which is the square of a function of the angle, called the amplitude.
Wave-particle duality: abstracting from rotation matrices (already covered) we get complex numbers, which provide a description of particles necessarily assigning them periods detectable by a two-slit experiment. The complex amplitudes distinguish two types of particle, the building-blocks of matter (fermions) from the constituents of radiation (bosons). Wave-like interference obliges light (and particles) to travel in straight lines. The complements to time and space needed to specify the waves turn out to describe energy and momentum, leading to E = mc^2 and the calculations of energies of fission and fusion.
Spin-1/2: because reflections are more fundamental than rotations, and because it takes two reflections, at half the angle, to make a rotation, spin-1/2 is an intrinsic property of three-dimensional space. (In two dimensions everything commutes, so rotations suffice.)
Antimatter: E = mc^2 says that particles can be created and destroyed. Abstracting from reflection matrices (already covered) gives Clifford numbers which take us beyond the two dimensions of complex numbers, and introduce us to the weird anticommutating product f1f2 = -f2f1. Such a product must apply to an operator which creates or destroys fermions (while bosons appear to follow the more conventional result b1b2 = b2b1). At the smallest, for relativistic fermions, the matrices to support this must be 4-by-4, leading to the discovery of antimatter.
Physics for Teenagers: Collective Phenomena
Synopsis
Individual particles have momentum and energy; their collectivity has pressure and temperature, both being averages. More subtly they have equilibrium, reflecting their distribution of energies, and heat, which is not itself a state of the collectivity but changes of which are work and give us heat engines and heat pumps. In what becomes the Boltzmann distribution, equilibrium maximizes energy-ignorance, which becomes the physical quantity entropy. Phase transitions involve all-scales phenomena of the collectivity and give rise to power-law distributions. Closer to equilibrium are transport phenomena, in which free-particle acceleration is restrained, by a filter provided by the equilibrium of the rest of the collectivity, to a terminal velocity. Turbulence is another all-scales phenomenon. Finally, life itself is a collective phenomenon situated in a possible phase transition between order and chaos.To introduce all this in detail to people whose mathematics is limited to simple algebra - and that a little shakey - requires basic introductions to sums and related areas (for the averages and, more generally, moments), to histograms and related distributions and probabilities, to the mathematics of small changes, and of course to momentum and energy and their great conservation laws. The book starts slowly but accelerates, so beginners reading later chapters may need support of an experienced mentor - but skimming the rest after some grounding in the first three chapters will also be rewarding.
We introduce summing via triangular and tetrahedral numbers and square and cubic numbers. Sums and areas approximate each other and areas can be easier to calculate. This leads to averages (which we compare with other measures of central tendency) and moments. Histograms allow us to work in the absence of detailed knowledge, are themselves characterized by averages and other moments, and lead to the probability distributions we'll be needing.
With this and the physics of momentum and energy, which we introduce for colliding particles, we can calculate pressure and temperature, and trace ideal gases in pressure-volume diagrams where areas measure energy output or input.
Small changes in a curve are captured by the slopes of the curve, and from that idea we find out about two special curves, the exponential and its inverse, the logarithmic. All this we can apply to and illustrate with van der Waals gases, a refinement of ideal gases which allows for phase transitions. It also leads to the probabilistic notions of surprisal and ignorance, using a simple non-physics fable.
That's the first three chapters. The remaining chapters go on to examine equilibrium and entropy, phase transitions, transport phenomena, turbulence, and, briefly from these points of view, life.
Audience: people who remember some algebra and are willing to learn about sums, areas, averages, probabilities, slopes, energy and momentum en route to the physics of entropy, resistive transport and phase transitions.
Under the hood
Entropy: a statistical distribution uses probabilities to cover our ignorance of detailed events. Entropy characterizes our ignorance even of the probabilities. Those probabilities refer to energies of molecules in a gas, for instance, and entropy can be measured in terms of heat (internal energy) and temperature. So it is a physical, not merely subjective, quantity. Entropy is maximized when the gas is in equilibrium, leading to the "second law of thermodynamics" that entropy never decreases, hence giving a direction to time not contained in the physics of the individual molecules ("emergent"). A fantasy of Dad's misplaced tools, and Mom's later reconstruction of the distribution, provides the derivation of the Boltzmann distribution that results when ignorance of the probabilities is maximized.Transport: multiplying two algebraic expressions, e.g., (a+b)*(a2 + 2ab + b2)
| | | a2 | 2ab | b2 | |
| --- | | | ------ | ------ | ------ |
| a | | | a3 | 2a2b | ab2 |
| b | | | a2b | 2ab2 | b3 |
Phase change and turbulence introduce activity at all scales, far from equilibrium, hence requiring not Boltzmann but power-law distributions, whose moments blow up - for instance the averages that give rise to thermodynamic properties increase without limit. For phase change, compressibility is one such average. For turbulence, the transport of momentum (relating viscosity to diffusion and mobility) becomes chaotic beyond certain flow velocities.
Physics for Teenagers: Rocket Science
Synopsis
The first four chapters address the technicalities of motion in space: how to move with nothing to push against, and how "motion" in space really means changing orbits, and that means changing velocity. The first is accomplished by conservation of momentum, one of the great principles of Newtonian physics, when rockets eject propellant and when "sails" reflect or absorb it. For rockets this says that the velocity change ("delta-vee") is proportional to the velocity of the propellant multiplied by a weak dependence on the amount of propellant carried. The second uses conservation of energy and conservation of angular momentum - more great physics - to characterize orbits, and discovers the delta-vee needed to change orbits via a transfer orbit.Energy is also the basis for discussing what kinds of source - chemical, solar, nuclear - can generate various levels of propellant velocity, for assessing how far out in the Solar System we can get (or even leaving it), and, later, for guessing when humanity will be capable of making these various journeys.
Our discussion of orbits also goes beyond possible voyages, to discover the weights (masses, really) of Earth and Sun, to see the effect of gravitational resonances on satellites and especially on the Asteroid Belt and Earth-crossing asteroids, to understand tides, and to find the special orbits called the Lagrange points.
Two more chapters explore the space adventure itself: when we'll get to the various "landmarks", how we can protect ourselves in the hostile environment (space debris near Earth, radiation, and no gravity), cheap (but advanced) ways of climbing out of Earth's "gravitational well", the biological (nutrients, genetics) and social issues of long-term space travel (interstellar!), alternatives to going there ourselves (self-reproducing machines), and why nobody else has come to us (so far) (radio-astronomy).
Finally we note that Earth itself is a spaceship, travelling faster (around the Sun, around the Galaxy) than any vehicle we've so far launched, but also in danger from the cosmos, not to mention (we do in some detail: more physics) from ourselves.
For the unfamiliar, or post-familiar, an appendix introduces the math needed: algebra, trigonometry and calculus.
Audience: people who remember some algebra and are willing to learn about momentum and energy and their conservation laws, as well as accept some results from calculus, en route to propulsion and orbits in space, and to semi-quantitative discussions of our future in space and on Earth.
Under the hood
Delta-vee: just as conservation of momentum and energy are fundamental to the relevant physics, ΔV, change in velocity, is significant for space travel. Conserving momentum gives the rocket equation which relates ΔV strongly to the exhaust velocity of the rocket (hence what type of rocket it is: chemical, nuclear, etc.) and weakly but significantly to the ratio of fuel to empty rocket. Conserving angular momentum and energy gives the vis-viva equation for velocities in elliptical orbits, hence the ΔV needed to change orbits around star or planet. Our current exploration of the solar system is based on ΔV and, of course, is highly quantitative.Questions. Do we need rockets to travel in space? What about sails? And elevators to get us up there? How and when might we reach the stars? What are the dangers? Where are the aliens? What about "Spaceship Earth": what has carbon-dioxide warming done in the past? Is there a carbon-dioxide explanation for the "little ice age"?
Computing for Teenagers
Synopsis
We introduce programming using Python on triangular numbers and binomial coefficients, and leading to functional programming including recursion. We elaborate with states and instances (generally known as object-oriented programming) in order to illustrate the idea of a universal computer by mimicking the programming language Logo with the programming language Python. We focus on the "turtle graphics" aspect of Logo, for which we need the mathematics of directions and components. We introduce also the mathematics of two-dimensional numbers, which gives an alternative version of turtle graphics and we use it to display regular polygons. Because the turtle winds up in its starting position and orientation in these figures we can interpret them as higher-dimensional vectors and learn a little - optionally - about Fourier series.With programming under our belts, the third chapter is a crash course in computer science: finite-state automata for a variety of applications, stacks and queues, pushdown automata for grammar recognition and processing, and searching (sequential, logarithmic and direct). Automata are introduced to convert Roman to Arabic numerals, used in simple form (but in two dimensions) for John Horton Conway's "game of life", and more fully for Chris Langton's self-reproducing machine. Both railway shunting and Edsger Dijkstra's shunting algorithm to convert an expression to "reverse Polish" for evaluation use stacks and queues, and so does a pushdown automaton to help parse grammars.
One of the joys of computing is exploring (simulating) mathematics and nature where the formalisms have not been worked out, and might not be able to be worked out. Chapter four looks at fractals from various points of view - bit interleaving, recursion and iterated function systems both deterministic and random. Nonlinear functions of 2-D and 1-D numbers lead to unending patterns, and to chaos in which miniscule changes in starting conditions give enormous variations in results.
Computers are based on George Boole's mathematics of logic (which he called the laws of thought), both "combinational" for central processing units, and "sequential" for memory. We build a multi-bit adder and a nand flipflop for one bit of memory.
Gaussian elimination to solve sets of linear equations can get into trouble, even if your program has no bugs, because computers store numbers to only some maximum of digits or bits. Other algorithms take too long to run for all but minimal inputs because their "complexity" is exponential or worse. There is a whole class of problems which appears to be exponential but computer science does not (yet) know: we look at boolean satisfiability. Some computational problems are even undecidable.
AI (artificial intelligence) has recently made a breakthrough and we finish with artificial neurons, both discrete (perceptrons, which can mimic any computer) and continuous (sigmoid neurons, which can learn by adjusting their parameters). Because AI depends on enormous computations and enormous training data we cannot go too far but we can, with a few wobbles, teach neurons to be simple logic gates.
Audience: people who remember some algebra and are willing to learn about programming in a particular language en route to basic computer science and engineering with applications including self-reproducing automata, fractals and artificial intelligence.
Under the hood
Computer programs are machines with no physical parts. Thus "neither snow nor rain nor heat nor gloom of night stays" them "from the swift completion of their appointed rounds" but they can become arbitrarily complex. Good techniques are necessary.Recursion: triangular and simplex numbers; factorials; greatest common divisor; grammars; fractals, leading to iterated function systems and chaos.
State and instantiation: counting sheep; turtle graphics.
Automata: Roman numerals; "game of life"; self-reproducing machines; grammars.
Stacks and queues: doing arithmetic.
Complexity and program design: gaussian elimination.
Neural networks: learning machines and A.I.
Building a computer: boolean circuits.
There are some programs not worth banging your head on: errors and intractability.
Advanced Math for Preteens
Synopsis
In the guise of becoming friends with numbers and all their myriad characters - triangular, tetrahedral, square, cubic, powers, perfect, prime, fibonacci - we painlessly introduce binomial coefficients, algebra, functions, quadrature, higher dimensions, number bases, rational and irrational numbers, limits, and quadratic equations. This is a Socratic dialogue.For triangles, squares, cubes and tetrahedra of dots we discover two types of rule to calculate the numbers of dots: Rule 1 is iterative, depending on the previous shape; Rule 2 is direct, given only by the number of dots per side.
The Rules 2 allow us to plot linear, quadratic and cubic functions, to extrapolate to whole numbers that go beyond the dot-pictures, and even to fractional numbers.
Then we find that adding up consecutive numbers, and adding up those sums, is closely related to finding areas under the plots and we discover ways of doing that exactly and approximately.
Triangles and squares extend in higher dimensions to simplices and hypercubes and we find our earlier numbers reappear in counts of their vertices, edges, faces, and so on.
Hypercubes require exponentiation and present an opportunity to be quantitative: we explore the very large and the very small in powers of 2^10 and 10^3, going from quarks to the Universe in fourteen steps.
Powers of 2, as compared with powers of 10, lead us to binary arithmetic (and computing) and we learn to convert between bases. Fractions in different bases motivate decimal notation and the notion of rational numbers.
Numbers that are the sum of their own divisors introduce prime and Mersenne numbers, and illustrate the risks of leaping to conclusions about apparent patterns.
Attempting a Rule 2 for Fibonacci numbers introduces limits, irrational numbers, and quadratic equations.
We conclude with a tip of the hat to Ramanujan and his friends.
Audience: people who are willing to learn about algebra, making pictures of algebraic rules, basic calculus, quantities from quarks to the Universe, binary arithmetic and other bases, and quadratic equations.