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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 |
requires summing numbers along top-right to bottom-left diagonals to get
a3 + 3a2b + 3ab2 + b3. A similar
diagonal-sum process (the "convolution")
is used in digital signal processing to filter one series according to another
series. For instance, a force applied to a charged particle in a neutral gas
can be filtered so as not to accelerate continuously but reach a terminal
velocity. A related process, but summing top-left to bottom-right diagonals,
gives the correlation between two series, such as the velocities of molecules
in Bolzmann equilibrium. If the two series are the same we have the
autocorelation, which can be used to filter out noise. It can also be used as
the force-filter above, and this is what happens in such cases as the transport
of charge through a material. In this case we get Ohm's law for electrical
resistance. In other cases we get mobility, diffusion, and combined transport
phenomena such as thermocouples.
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.
Under the hood
Numbers have personalities. They can be even or odd. They can be composite or
prime. (Can a prime number be even?) They can be deficient. abundant or perfect.
(How do perfect numbers relate to primes?) They can be powers. (How do perfect
numbers relate to powers of 2? What is the connection between powers of 2 and
powers of 10? How can we go from quarks to the Universe in 15 steps? How can
we multiply by adding?) Some numbers give the family trees of bees. And they
can have shapes: triangular numbers, square numbers, hexagonal numbers, cubic
numbers, tetrahedral numbers. Sometimes we have rules for these personalities,
sometimes not. When we have rules we can draw them and even find areas in the
drawings. And we can count on two (or three) fingers instead of ten: it's
easier.