Where to go to get lost :-)

Highly recommended!
(Be sure to audition the crystalline soundtrack!)
For its home web page, click here.
(Its Wikipedia page.)

(Note: Once it starts playing,
both the video and audio are considerably sharper
if you click on the lower right “360p”, changing it to “480p”.
(On my Microsoft system, in both Firefox and IE, anyhow.))


















The 120-cell {5,3,3}:

The 600-cell {3,3,5}:

Finally, a mere 24-cell {3,4,3}:

---

But wait; is all that too boringly static?
Then watch these convex regular 4-polytopes rotate for you:
(Forget “Dancing with the Stars”.
Try “Twirling with the Topes”, or “Pirouetting with the Polys”!)

4-Simplex or 5-Cell {3,3,3}	4-Cube or 8-Cell {4,3,3}	4-Cocube or 16-Cell {3,3,4}
24-Cell {3,4,3}	120-Cell {5,3,3}	600-Cell {3,3,5}

Stepping down a dimension, here are the five Platonic solids,
i.e., the five convex regular polyhedra in three dimensions.

Tetrahedron or 3-Simplex {3,3}	3-Cube {4,3}	Octahedron or 3-Cocube {3,4}
Dodecahedron {5,3} Icosahedron {3,5}

How’s that for a nice break from work :-)
Sorry I haven’t found rotating views of the two biggies —
the 120-cell {5,3,3} and 600-cell {3,3,5}.
You might note that, for the 4-polytopes,
the two polytopes on the left are each self-dual,
while in each row the two right-hand polytopes are dual one to the other.
For the 3-polytopes, the {3,3} is self-dual,
while the other polytopes in each row are dual one to the other.

You can check out all six convex regular 4-polytopes,
or take a look at all sixteen (not-necessarily-convex) regular polychora —
there is some very interesting Java animation at that site
(in some cases you have to move your mouse onto the figure to see it).
For the Platonic solids, see here.

By the way, if your display doesn’t have enough vertical height
to display all the polytopes without scrolling,
at least on some Windows systems the F11 key will expand the vertical scope.
Interestingly, in my system, the polytopes spin faster
in Firefox than in Internet Explorer.

---

Thanks, Wikipedia!

These are hardly the only ways of
trying to depict these four-dimensional figures in two dimensions,
a difficult task at best.
There are many others, each with its own advantages and disadvantages.
The Schlegel wireframes above are dramatic and perhaps aesthetic,
but that isn’t everything.

A fascinating, thorough, prize-winning article is
“The Story of the 120-Cell” by master expositor John Stillwell.
You also might want to check out John Baez at Week 155.

You can also see what Google turns up for the polytopes:
24 {3,4,3}, 120 {5,3,3}, 600 {3,3,5}.

Bashing the "new math"

Why Do Americans Stink at Math?
By ELIZABETH GREEN

What math should be taught?

2015-12-03-NYT-Phillips-the-politics-of-math-education

The Politics of Math Education
By CHRISTOPHER J. PHILLIPS
New York Times Op-Ed, 2015-12-03

Pittsburgh —

[1]
AMERICAN children have been bad at math for well over a century now. As early as 1895, educational reformers lamented Americans’ “meager results” in the subject. Over the years, critics of math education in this country have cycled through a set of familiar culprits, blaming inadequate teacher training, lackluster student motivation and faulty curricular design. Today’s debates over the Common Core mathematical standards are just the latest iteration of this dispute.

[2]
Although these issues are important — no reform can ever succeed without considering teacher training and textbook design — resolving them will never make the underlying question of how to teach math “go away.” This is because debates about learning mathematics are debates about how educated citizens should think generally. Whether it is taught as a collection of facts, as a set of problem-solving heuristics or as a model of logical deduction, learning math counts as learning to reason. That is, in effect, a political matter, and therefore inherently contestable. Reasonable people can and will disagree about it.

[3]
Perhaps no reform has illustrated this point as clearly as the wide range of mid-20th-century curricular changes known as the new math. Many of these reforms promised that the introduction of sets, nondecimal bases and formal definitions would lead students to think of math as more than just a bunch of dusty facts: It was a powerful and rigorous way of approaching complex problems.

[4]
The new math was widely praised at first as a model bipartisan reform effort. It was developed in the 1950s as part of the “Cold War of the classrooms,” and the resulting textbooks were most widely disseminated in the 1960s, with liberals and academic elites promoting it as a central component of education for the modern world. The United States Chamber of Commerce and political conservatives also praised federal support of curriculum reforms like the new math, in part because these reforms were led by mathematicians, not so-called progressive educators.

[5]
By the 1970s, however, conservative critics claimed the reforms had replaced rigorous mathematics with useless abstractions, a curriculum of “frills,” in the words of Congressman John M. Ashbrook, Republican of Ohio. States quickly beat a retreat from new math in the mid-1970s and though the material never totally disappeared from the curriculum, by the end of the decade the label “new math” had become toxic to many publishers and districts.

[6]
Though critics of the new math often used reports of declining test scores to justify their stance, studies routinely showed mixed test score trends. What had really changed were attitudes toward elite knowledge, as well as levels of trust in federal initiatives that reached into traditionally local domains. That is, the politics had changed.

[7]
Whereas many conservatives in 1958 felt that the sensible thing to do was to put elite academic mathematicians in charge of the school curriculum, by 1978 the conservative thing to do was to restore the math curriculum to local control and emphasize tradition — to go “back to basics.” This was a claim both about who controlled intellectual training and about what forms of mental discipline should be promoted. The idea that the complex problems students would face required training in the flexible, creative mathematics of elite practitioners was replaced by claims that modern students needed grounding in memorization, militaristic discipline and rapid recall of arithmetic facts.

[8]
The fate of the new math suggests that much of today’s debate about the Common Core’s mathematics reforms may be misplaced. Both proponents and critics of the Common Core’s promise to promote “adaptive reasoning” alongside “procedural fluency” are engaged in this long tradition of disagreements about the math curriculum. These controversies are unlikely to be resolved, because there’s not one right approach to how we should train students to think.

[9]
We need to get away from the idea that math education is only a matter of selecting the right textbook and finding good teachers (though of course those remain very important). The new math’s reception was fundamentally shaped by Americans’ trust in federal initiatives and elite experts, their demands for local control and their beliefs about the skills citizens needed to face the problems of the modern world. Today these same political concerns will ultimately determine the future of the Common Core.

[10]
As long as learning math counts as learning to think, the fortunes of any math curriculum will almost certainly be closely tied to claims about what constitutes rigorous thought — and who gets to decide.


[I (the author of this blog) went to high school in the early 1960s,
majored in math in college in the mid 1960s,
and studied math as a graduate student in the late 1960s and early 1970s.
I was steeped in the ideas of the "new math".
I thought it was an excellent approach.
My only complaint, which may reflect my own shortcomings,
was that what was taught often seemed too abstract,
without examples.
For example,
somehow I lacked (maybe it was taught but I ignored it)
a thorough grounding in the group of isometries of the Euclidean plane (and 3-space),
showing the relations between reflections, rotations, translations, and glide reflections
and the subgroups that they generate.
Such would provide an ideal (in my opinion) introduction to group theory,
together with enhancing ones understanding of geometry.
See, for example, the Springer UTM texts
Martin, George E. (1982). Transformation Geometry: An Introduction to Symmetry. ISBN 978-0-387-90636-2.
Armstrong, M. A. (1988). Groups and Symmetry. ISBN 978-0-387-96675-5.
This subject shows how the ideas of the "new math"
(in this case, group theory which was introduced in the 19th century)
can be used to analyze Platonic solids,

which surely are among the most classical ideas in mathematics.
An interesting, in my opinion, combination.]

Elementary mathematics

Computing the compounding rate

A common problem is computing the rate of compounding,
given the initial amount, final amount, and number of periods.
We start with an initial amount a.
It increases for n periods;
in each period the amount then current
is multiplied by a constant factor r
to obtain the amount for the next period.
The final amount is b.
The formula that results from this situation
is the compounding or exponential growth formula

b = a × rⁿ .

It frequently happens that
we know
the initial amount a, the final amount b, the number of periods n,
and wish to determine
what multiplier r
would have yielded that final amount
when compounded as described.

To solve this problem we need only the following facts:

exp and log (aka ln) are inverse functions
exp ( x + y ) = (exp x) × (exp y).

From those two facts it easily follows that

log ( a × b ) = (log a) + (log b)
log ( b/a ) = (log b) - (log a)
and
log ( rⁿ ) = n × log r.

[Pedantic note 1:
So the exponential is a homomorphism, in fact an isomorphism,
from the (abelian group of) reals under addition and subtraction
to the (abelian group of) positive reals under multiplication and division,
while its inverse the logarithm is a homomorphism, in fact an isomorphism,
in the opposite direction.]

Now
to solve the problem of computing the rate of compounding as stated above,
we observe the following string of logically equivalent equalities:

b = a × rⁿ
b/a = rⁿ
log (b/a) = log ( rⁿ ) = n × log r
log r = ( log (b/a) ) / n
r = exp ( ( log (b/a) ) / n )

(The second line above is equivalent to r being the nth root of b/a, r = (b/a)^1/n,
but to actually compute the nth root you’re back to working with log and exp.)

For an application of the last formula,
see the calculation of the annualized rate of tuition increase at Harvard here.

Pedantic note 2:
The key part of the final formula for the rate of compounding,
exp ( log (??) / n ),
is an example of conjugation,
generalized from the theory of groups to the theory of groupoids;
the groupoid in which this conjugation takes place is the
groupoid of (abelian) groups and isomorphisms.

This conjugated function may be represented as a simple diagram
in the groupoid of (abelian) groups and isomorphisms:

log (R+, ×, 1) ------------------> (R, +, 0) | | ?/n | V (R+, ×, 1) <------------------ (R, +, 0) exp

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Miscellaneous Articles

2010-03-06-NYT-Bayley-Algebra-in-Wonderland

Algebra in Wonderland
By MELANIE BAYLEY
New York Times Op-Ed, 2010-03-06

Oxford, England

[1]
SINCE “Alice’s Adventures in Wonderland” was published, in 1865, scholars have noted how its characters are based on real people in the life of its author, Charles Dodgson, who wrote under the name Lewis Carroll. Alice is Alice Pleasance Liddell, the daughter of an Oxford dean; the Lory and Eaglet are Alice’s sisters Lorina and Edith; Dodgson himself, a stutterer, is the Dodo (“Do-Do-Dodgson”).

[2]
But Alice’s adventures with the Caterpillar, the Mad Hatter, the Cheshire Cat and so on have often been assumed to be based purely on wild imagination. Just fantastical tales for children — and, as such, ideal material for the fanciful movie director Tim Burton, whose “Alice in Wonderland” opened on Friday.

[3]
Yet Dodgson most likely had real models for the strange happenings in Wonderland, too. He was a tutor in mathematics at Christ Church, Oxford, and Alice’s search for a beautiful garden can be neatly interpreted as a mishmash of satire directed at the advances taking place in Dodgson’s field.

[4]
In the mid-19th century, mathematics was rapidly blossoming into what it is today: a finely honed language for describing the conceptual relations between things. Dodgson found the radical new math illogical and lacking in intellectual rigor. In “Alice,” he attacked some of the new ideas as nonsense — using a technique familiar from Euclid’s proofs, reductio ad absurdum, where the validity of an idea is tested by taking its premises to their logical extreme.

...

Neat pics

Cf.; to rotate it click here and use your mouse (hold down the left key).

More



You may wonder what to call
the figure shown in the top row
in various stages of undress.
Well, if you like Latin
you may call it the “stella octangula”;
if you prefer English
try “stellated octahedron”;
and if you like a systematic naming scheme
that gives
lots of numerical information about it
and admits many useful generalizations
(including to arbitrarily high dimension;
it is really a very powerful notation),
try “{4,3}[2{3,3}]{3,4}”.
The last is the version for compounds
of notation due to Herr Schläfli,
and is therefore called a
“Schläfli symbol”.
What those numbers signify, roughly, in English is
a cube {4,3} containing (circumscribed around)
two reciprocal (i.e., dual) tetrahedra {3,3} which intersect in
an octahedron {3,4}
(so three of the five Platonic solids appear in it).
Conglomerates of this nature are generally called compounds,
more specifically, polyhedral compounds.





Here is a big brother of the
{4,3}[2{3,3}]{3,4} cube/tetrahedra/octahedron compound:


This consists of
an outside dodecahedron {5,3}
whose 20 vertices are also the vertices of
five tetrahedra {3,3}s which intersect in an
icosahedron {3,5}.
Its Schläfli symbol is, fairly naturally,
{5,3}[5{3,3}]{3,5}.
In this figure, each of the five tetrahedra has its own color,
red, blue, light blue, green, or brown.
The edges of each tetrahedron are in that color,
as well as
the faces of the icosahedron which arise from that tetrahedron.

Here is what the five tetrahedra look like,
without the surrounding (circumscribed) dodecahedron
(that is, the compound of five tetrahedra),
first as (colored) solids, then as thickened outlines:



And here is the compound being built up,
starting with the core icosahedron and adding one (or a few) tetrahedra at a time:



You will find everything you ever wanted to know about such things
in Regular Polytopes
by H. S. M. Coxeter
(a real classic, first published in 1947;
it has its own Wikipedia entry).
The Schläfli symbols for compounds (like the above)
are explained in Section 3.6.


The illustrations above are a selection of those from
Glimpses of Algebra and Geometry by Gabor Toth.
Only about fifteen percent of that book is devoted to such solids,
but the approach there seems more user-friendly than Coxeter.







Here is a question:
If the twenty vertices of the dodecahedron {5,3} can be divided into
the vertices of five tetrahedrons (each tetrahedron {3,3} having four vertices),
what can be done about dividing them into the vertices of cubes {4,3}?
Well, it turns out they can be divided also into the vertices of five cubes.
Since each cube has eight vertices, making a total of forty cube vertices,
each dodecahedron vertex is the vertex of two cubes.
Here is the picture:



To see the cubes, note that the white pentagonal face of the dodecahedron
has five diagonals (making a five-pointed star), each colored a different color.
Those five diagonals are edges of the five cubes.
In fact, the same is true of each of the twelve faces of the dodecahedron:
Each has five diagonals, making a total of sixty diagonals.
But since each inscribed cube has twelve edges,
the five cubes also have sixty edges.
How about that!
The Schläfli symbol for this is
2{5,3}[5{4,3}].
The {5,3} denotes the outer dodecahedron,
the 5{4,3} denotes the five inscribed cubes,
while the 2 prefixing the {5,3} denotes, not that there are two of them,
but that each vertex of the {5,3} is also the vertex of two {4,3}s.
Another way of viewing this is that it gives the following (valid) equation:
2 × (20 vertices per {5,3}) = 40 = (5{4,3}) × (8 vertices per {4,3}).
Again, to read about the Schläfli symbols for compounds such as this,
see Section 3.6 of Coxeter.

Oh, and here are two representations of the
compound of five cubes,
that is, the 5{4,3} compound without the circumscribing {5,3}.
The one on the left is by Toth,
and colors the five cubes red, two shades of blue, green, brown;
the one on the right is by George Hart,
and colors the five cubes red, blue, green, yellow, brown.


Note how each vertex of the {5,3} dodecahedron has a pair of colors.
There are 5 × 4 = 20 possible ordered pairs (without repetition)
of the five colors.
Each such ordered pair appears on exactly one
of the 20 vertices of the {5,3} dodecahedron.
Reversing the order of the colors corresponds to going to the antipodal vertex;
e.g., in the right hand picture
the blue/green vertex is the antipode of the green/blue vertex.

---

Web Sites

Here are some web sites that look interesting:
polyhedra.org
“The Encyclopedia of Polyhedra” by George W. Hart
“A Ride Through the Polyhedra World” by Maurice Starck
“Platonic Solids in All Dimensions” by John Baez

or just see what Google finds for:
(polyhedra|polyhedron|polytope)
regular polyhedra

Geometry

Euler’s classification (1776) of isometries in dimensions ≤ 3.

Euler’s classification (1776)
of isometries in dimensions ≤ 3

Isometry Eucl	Fixed Point?
	Yes	No
OP	rotations ρ (dim ≥ 2)	translations τ (dim ≥ 1) screw (dim ≥ 3)
OR	reflections σ (dim ≥ 1) rotary reflections (dim ≥ 3)	glide reflections γ (dim ≥ 2)

Notes:

1. OP = orientation-preserving = even = direct = Isom⁺
   OR = orientation-reversing = odd =opposite = Isom^-


2. Each non-identity isometry is in precisely one of the above categories.


3. Wikipedia discusses:
   Classification of Euclidean plane isometries and
   Overview of isometries in up to three dimensions.


4. Reflections
   More precisely, we mean reflections in a hyperplane of codimension 1
   (thus in 2-space, in a line, in 3-space, in a plane)
   with respect to the orthogonal line to the hyperplane.


5. Rotations in 2-space are with respect to a point,
   in 3-space with respect to a line.
   Thus rotations are always within a plane,
   about the orthogonal complement.
   Of course, in dimensions ≥ 4 things get more complicated,
   and are best discussed in terms of
   the decomposition of orthogonal operators studied in linear algebra.
   This link, if it is still active, gives the relevant theorem.
   There is also Wikipedia’s discussion of the orthogonal group,
   but that covers much more than is immediately relevant.


6. Glide reflections:
   The direction of the glide is always within the reflecting hyperplane:
   in 2-space the reflecting line, in 3-space the reflecting plane.
   Glide reflections in 3-space are sometimes called
   “glide plane operations”.


7. Screw = rotary translation = glide rotation


8. Inversions
   By inversion
   we mean what is sometimes called
   “central inversion” or “inversion in a point.”
   In 2-space, an inversion through a point
   is just a half-turn about the point, thus a rotation.
   In 3-space, for inversion through a point
   we may choose any line through that point,
   perform a half-turn rotation around that line
   combined with a reflection in
   the plane orthogonal to that line and through that point,
   thus we have a rotary reflection.
   This example shows that although each isometry
   is in one and only one of the above categories,
   its precise expression within that category is not unique.


9. The familiar (from high school geometry) property of
   congruence of figures in the plane
   simply means that
   one figure is the image of the other under an isometry.


10. There is doubt as to how much of that classification
    Euler really discovered.
    I’m giving him the benefit of the doubt,
    following Coolidge’s History of Geometrical Methods
    as cited in George Martin’s Transformation Geometry.