Should I Become a Mathematician?

[musicheck]

Mathwonk, have you felt that your new insights into classical geometry have helped you understand modern (say differential or algebraic or something) geometry?

 

[mathwonk]

i did answer but the browser trashed it. basically it helps me focus on what is important to learn, and i hope to learn to do that for more modern topics.

i.e. classical math has already helped me understand calculus better. so next i hope to move up to power series and then analysis and deformation theory, algebraic geometry,...

i am just trying to understand what to focus on and how to use what came before to understand what came next.

 

[Phred101.2]

Have you developed a good model (understanding) of harmonic motion, that we see so much of everywhere? Do you understand Euler's thms. or any of Godel's stuff?

 

[mathwonk]

no. not much. i learn in very small steps. which of eulers theorems do you refer to? he has hundreds of pages of collected works. i am beginning to appreciate his amazing use of power series and infinite products.

godel is over my head. i am not really interested in logic at present. to me it is navel gazing, by people worried over the validity of what i enjoy! but hurkyl differs.

 

[quasar987]

Do you understand Euler's thms.

It's good you put an "s" there.

 

[mathwonk]

problem: use archimedes method of parallel slices, to show that the volume contained in the intersection of two perpendicular cylinders inscribed in a cube, is 2/3 that of the cube.

 

[quasar987]

Is this the method he used to show a sphere inscribed in a cylinder occupies 2/3 of the volume of the cylinder?

 

[pivoxa15]

the one fascinating course that went into interesting geometric material was Bott's algebraic topology, but regrettably I bailed from it early in the semester (as a senior) because I had so little background for it.

I nearly did the same that is bail out of algebraic topology (was your's an undergradute subject? I thought you said you did really well in (point set) topology?) early for the same reason but I asked your advice and you said not to and so did the lecturer. In the end it was good that I didn't mainly because now I know how hard maths can be. It's the only subject that gives me a headache.

Could you describe the very basics of algebraic geometry? How similar is it to algebraic topology?

 

[Phred101.2]

My sister knows this young guy who apparently figured out something about Euler's 2-ribbon thing. He's a savant, or something, he saw this paper about it on th'web and contacted the author to point out a problem he had spotted. He was the only other person, apart from the prof., to spot it, or something.
Anyhoo, Godel's incompleteness thm. and NP-complete solutions are part of the same set of "problems", if your into algorithms and computational theory. I'll go out on a limb here and say I have another sister with a degree in Anthopology...
(of course, I could also be just a pizza-guy)

 

[mathwonk]

oops quasar, i answered the wrong question. yes use the same method.

i conjecture that he solved this problem in an analogous way. when i read his statement that he had proved this, i immediately thought of a solution analogous to his earlier one. then i read that his solution was lost, but zeuthen (100 years ago) had given a plausible one that he might have given. zeuthens solution is essentially the same as mine, but phrased in terms of centers of gravity instead of volumes.

an outline sketch of the very basics of algebraic geometry are described in a short paper on my website i wrote for my conference. i also posted it here back in the spring, posts 491-497.

 

[mathwonk]

it turns out archimedes methods also give the surface area for a bicylinder, something seldom seen even in calculus classes, as well as both volume and area for a tricylinder too! it has recently been argued therefore that he knew at least the surface area of a bicylinder although he is not known to have claimed it.

 

[mathwonk]

by the way one thing this has taught me is how intimately physics insight is connected to mathematical proof and discovery. archimedes revealed the link between volume and centers of gravity, and i doubt if i will in future ever again omit centers of gravity from my discussion of volume!

for years i confess to have often skipped the grubby applied sections of the book, whereas archimedes used them to discover the idea of volumes by slices (cavalieri principle)!

why do we not all learn this in high school? teaching high schoolers rote calculus, instead of archimedes' and euclid's geometric ideas that underlie calculus, is academic stupidity bordering on insanity.

Although I did not have a decent physics class, I had both good high school geometry and algebra classes. This is probably why I survived my high level intro to college calculus, whereas my own students usually do not.

I conjecture a large part of the American deficiency in science and maths stems from the pitiful high school curriculum which omits large parts of geometry and algebra, focusing instead on a narrow minded attempt to prepare the students as quickly as possible for a shallow, formulaic AP calculus course.

I.e. the focus on teaching only to the questions that are usually tested on standardized tests, as opposed to teaching for understanding of the topics, makes it essentially impossible to advance far at all. I have for partly this reason spent my whole life reconstructing the root ideas of calculus.

There is no royal road to calculus, or any other subject. One must master geometry, then algebra, and basic concepts of physics are very central. In the theory of integration Riemann's role is far over stated. His definition, while precise, is quite inessential to understanding volumes and areas, and only makes the contribution of determining which functions are integrable, an aspect actually omitted in most courses, i.e., Riemann's contribution was to show that a function is integrable in his sense iff it has discontinuities only on a set of measure zero, which is generally thought due to Lebesgue.Essentially everything else we teach in a calculus course is due to Newton, and Archimedes, and is based in geometry, physics, and algebra. Even Euler's ideas get little attention, such as the interplay between infinite sums and infinite products.

Riemann's main contribution was in complex analysis, and its applications to number theory, and in differential geometry, where he gave the concept of manifolds and generalized Gauss' curvature. He then used manifolds to unify the study of complex functions by organizing them into families parametrized by "moduli", but little of this is taught to beginning students.remark: what i now see as the correct order of topics, i.e.classical insights first, including area, volume, with cavalieri's principle, is the order used in the book of apostol, although it is quite exacting and strenuous there. one could make it look easier too and still use the same approach. but apostol does things historically right, as few other books do. note also that courant does calculus in this order, integration first, then differentiation. this is the way it should be done, and was done, and the opposite approach is perhaps merely a concession to the modern desire to get to the end more quickly, unfortunately sacrificing understanding along the way.

spivak does things in the opposite, modern, order, in his lovely book, but one may argue that his calculus book is really a baby analysis book, with little attention to or interest in, the beautiful historical applications to area and volume and physics.

 

[muppet]

This thread is going wonderfully off topic
Does it shock you, mathwonk, that as a 2nd year undergraduate I'd never heard of Cavalieri's principle prior to this thread?
I've never seen a treatment of calculus that begun with integration and then covered differentiation. It's an interesting idea that doing it that way might help people gain a deeper understanding. Do you think that the change of emphasis might obscure the idea that you can approximate a function over a small region by its derivative?

 

[Gib Z]

On Muppets last line, do you mean by its tangent, which is computed with the aid of its derivative? And I've read about Cavalieri's principal, but at school my teacher only mentioned it in passing, and more sadly, when another student asked how to prove it, my teacher *proved* it by saying "Isn't it just obvious?". sigh

 

[muppet]

I do really mean tangent... I used the word derivative because when you formulate problems in physics the differentials you multiply by small changes aren't usually straight lines. Forgive my sloppiness on the maths thread!

 

[mathwonk]

well muppet, [ i sound like mrs. doubtfire], i believe archimedes used that idea to compute arclength and surface area.

i.e. i gather he approximated the length of a circle by the lengths of polygons made up of segments tangent to the circle, as well as by secants to the circle. He then showed those have the same limit.

moreover he computed the surface area by approximating it in strips, with each circular strip approximated by a conical strip generated by revolving a tangent segment to the circle which sweeps out the sphere.

i.e. a sphere is swept out by revolving a circle,

hence we approximate the surface of a sphere by revolving the tangential segments that approximate the circle.

and to me this beautiful stuff is not even off topic. i.e if you want to be a mathematician i suspect you cannot do better than to consider what is being said now about appreciating archimedes, original sources in general, and historical evolution ,of important concepts.

 

[mathwonk]

it is interesting to wonder just what the advance was from archimedes to Newton.

i mean archimedes already undertstood that the surface area of a sphere determined its volume and vice versa, which is basically the inverse relation ship between integral and derivative, so he actually DID have the FTC allowing him to go from the derivative, i,.e. the surface area, to the volume, its integral.

maybe it was just algebraic notation, something we take for granted. i.e. integrating formulas algebraically seems to be an advance. and arabic numerals are a big advance on roman ones.

i have not seen yet whether he understood that the relationship between area and height of a parabola, is the same one as between the surface area and volume of a sphere.

of course he solved them by the same approach!

i hate to accept somehow that as mundane a thing as notation is that crucial. but boy i sure hate the notation for tensors and sheaf cohomology. I'm sure they are a big reason those concepts are hard to grasp!

 

[Cincinnatus]

I think these are interesting questions, how much of our understanding of mathematics is dictated by the notation we use? How have the popularizations of various notations influenced the historical development of mathematics?

Is there a version of the Sapir-Whorf hypothesis from linguistics at play here? That is, the Sapir-Whorf hypothesis says that the natural language (its grammatical categories and such) shape the way we think about the world. Is the same true for mathematics, that our mathematical notation shapes the sorts of mathematical objects we might think about?

 

[mathwonk]

i think it was the rise of algebra that enhanced the ideas from geometry of the greeks, and the interplay of the two.

i can imagine mentally that the area of a slice is the derivative of the moving volume function. but to take full advantage of that fact it helps then to write down an algebraic formula for that area and take an antiderivative algebraically.

of course archimedes could already do that antiderivative (from R^2 to R^3/3) since it only amounts to multiplying R^2 by R/3.

so maybe the only advantage is in being able to do the same operation for solids that i cannot imagine, but whose equations i can write down. in calculus we find areas all the time by integration for regions we cannot really picture at all.

 

[uman]

I have a question.

I'm a high school student aspiring to become a professor. I'm interested in math, chemistry, and physics but also in history and philosophy.

What are the advantages, disadvantages, and unique things about being a mathematician as opposed to any other academic?

Also, how does one know if one is good enough at math to be a mathematician?

 

[Gib Z]

Also, how does one know if one is good enough at math to be a mathematician?

If you really enjoy the subject, you're good enough.

 

[mathwonk]

most mathematicians i know did not make the choice because of any advantages over other areas. they just loved it so much they wanted to be in the field.

we all have fears we are not good enough at various times. there are however certain measures that separate the wheat from the chaff. e.g. if you cannot get good enough marks to be accepted to grad achool, you will have trouble becoming a mathematician.

in grad school if you cannot pass the PhD prelims, you may have trouble finding an advisor. but mainly it is determined by whether you enjoy it and find reasonable success in courses.

i always thought i was talented at math as a kid, teachers said so, i won contests, and I found myself able to solve problems many others could not. but that is in the early stages.

later it is more a matter of applying yourself. there is a such a shortage of math majors at many schools, that the competition is very sparse, and lots of inducements are ofered to get people to choose math. even if they are not future fields medalists, they can still become mathematicians of some stripe.

we all find our own niche in a subject we enjoy and are committed to.

how talented do you have to be to be a movie actor? robert deniro is one, and robert duval, but pauly shore is also one, and sylvester stallone. gauss and riemann, archimedes and john nash, were mathematicians, but so am i.

 

[uman]

Thank you for the advice. I still have a long time to decide, but for the moment I'm passionate and interested in math. One step at a time, I guess :-)

 

[mathwonk]

Here is another opinionated remark generated by this experience with archimedes, on learning and teaching mathematics.

A course on a new topic of mathematical theory ought to begin with an examination of the problems that the theory will solve. Then one should generate methods and techniques suitable to that problem, and test them on it to see how well they suffice for the given purpose.

In particular a book or course for beginners should almost never begin with a bunch of definitions. A course like that assumes the reader already knows the purpose of what he is about to learn.

The reason math books and courses are so inaccessible to average persons is this phenomenon, and probably euclid deserves much of the blame for that. we try too often to present a completely finished, axiomatic treatment of a subject that has been gestating for years, maybe hundreds or thousands of years.

on the other hand i find euclids own book a wonderfully beautiful treatement of an elementary topic. so maybe one can get away with it, with minimal introduction, in a case where the objects being studied are very simple, and very familiar, like lines, circles, and triangles.

but to start someone out with the definition of a complex manifold, or a tensor bundle, or a connection, is a bit much for me. I think one should begin by saying what problems lead to the need for these theories and definitions, and how they will be useful.

You will see if you look, that my graduate algebra notes are written in this style, where Galois theory is presented as an attempt to explain why some polynomials have solutions formulas involving only radicals, and others of higher degree do not.

I.e. I took as my watchword, motivation, even more than clarity, as the ideal for a good textbook. "Learn all this, someday you will understand why", just does not work with me. For this reason I object strenuously to including topics in a course syllabus just because they will be needed in a later course, with no inclusion of the reasons they will be useful later.

As another example, I tried to motivate (commutative) tensors in a thread here as a tool for expressing higher Taylor polynomials of functions on manifolds. The reaction i got showed this was something many had never heard before, even those more conversant than I with tensor notation and manipulation.

In high school mathematics, the John Saxon books are some of the very worst offenders in offering mere manipulation, as opposed to motivation and understanding. Manipulation is fundamental, but almost useless without an ability to discern when the manipulations will be needed. If you are being home schooled by a well meaning parent using those books, ask them to read and consider these words.

This is one place where high school courses (in USA) are usually lacking since the teacher has so little training as not to have a clue why the material is offered except that it will be on some standardized test. This is another reason to study calculus in college.

If your courses or books do not provide this motivation, and you are lucky enough to have an actual live teacher, ask what the theories being presented will be good for, and if you even luckier, the professor will not only know the answer, but will take time to tell you.

 

[mathwonk]

i'm grading exams, and here's a piece of advice:

it is utter folly to walk into an exam covering a particular topic not knowing how to work even the examples of that topic that were actually worked out on the board in class.

few things frustrate a teacher more as to the hopefulness of his task, than to actually put on the exam THE SAME EXAMPLES WORKED ON THE BOARD AND IN THE BOOK, AND ON PREVIOUS TESTS, and still many people cannot do them.

As a freshman at harvard, i recall that after a disastrous first hour exam in honors advanced calculus, professor sternberg apparently gave the same exam again as the second hour exam, with much the same results for many students.

I guess many people simply do not care whether they learn the subject or not. This is a fact of life for all who plan to go into this line of work. don't let it destroy your love of your subject or your enjoyment of the positive aspects of your career.

After years of teaching I have learned that some students think that it is the professor's job to give a test that they can pass, so that if they do poorly enough, he will reduce the difficulty. there is indeed great pressure to do this today, but some professors at some places still believe it is the student's task to either master the material appropriate to the course he is in, or get out.

To the students wondering if they are smart enough to do math, in school at least, most of it is not a matter of being super intelligent, simply a matter of doing the basic expected work which is spelled out. In most classes if you even do that you will shine above many many others for a very long time.

 

[Tickitata]

how do graduate schools view minors? would an adcom prefer a student who fills his electives with extra math courses, or a student who obtains a minor (say, in philosophy or physics) while still excelling in math?

 

[musicheck]

Out of curiosity, does anyone know if there are English translations of any of the Indian work on calculus? I read somewhere that in the 12th century they figured out differentiation, Taylor series, and even a bit of analytic number theory.

 

[quasar987]

Just a fun coincidence: I was just over at the math departement of my uni and saw on the billboard a nice picture of a flower plant with the title "Algebraic geometry and Varieties something conference in the honour of Roy Smith's 65th birthday".

Did you enjoy the party?

 

[mathwonk]

yes indeedy. it was the biggest professional honor of my lifetime. you may notice that the speakers included the head of the Institute for Advanced Study in Princeton, and numerous other wonderful mathematicians from USA and abroad, some lifelong friends and some new acquaintances, well known lights and rising stars. This was a conference put on by two of the outstanding young people in my department, Valery Alexeev and Elham Izadi, and they dedicated it to me, as a thoughtful kindness.

I was on a week long high that has not entirely dissipated. It is a wonderful treat to have the best people come in and tell you their latest work in your field, especially for your birthday. I recommend it to all. It was so inspiring that right afterward, Robert Varley and I reproved a small but interesting result due to one of the speakers, by a different method beloved by us.

 

[mathwonk]

in math we care practically nothing at all for minors certainly not in philosophy. its all pretty much math, although a physics minor might count for something.

 

[mathwonk]

calc 2 final

Try this on for size, all high school AP students, planning to exempt calc 1:

2260 Final Fall 2007 NAME:
No calculators, phones, books, notes,…; Show all work for full credit.
I.a. Give Riemann’s definition of the integral of a function f over an interval [a,b], as a limit of “Riemann sums”, and explain the meaning of the symbols.
b. Give two “independent” properties of f (neither one implies the other), each of which guarantees existence of the limit above.
c. Which property of a function allows antidifferentiation to (theoretically) be used to calculate the limit defining the integral, according to the fundamental theorem?
d. When f is defined by the rules below, explain why f is integrable [0,1], and yet the fundamental theorem is of no use in computing it:
f(x) = 1/3 for x in the interval [0 2/3],
f(x) = 1/9 on [2/3, 8/9),
f(x) = 1/27 on [8/9, 26/27), etc…
(i.e. f(x) = 1/3^k, on [(3^(k-1) -1)/3^(k-1), (3^k -1)/3^k), for all k ≥1,
and f(1) = 0.
e. Compute the integral in d. precisely, by summing the series of areas of rectangles formed by its graph.

II. Use the fundamental theorem of calculus to find the area of the plane region bounded by the curves y = ln(x^x), y = ln(1/x^e), and x = e. Why does it apply here?
(First sketch the graphs of these curves, finding where they meet.)

III. Using any method you know, (say which method you use),
Either:A) compute the volume of the solid formed by revolving around the y axis, the ring shaped region (“annulus”) formed by removing the circle of radius 1 from the circle of radius 2, both centered at the point (5,0);
Or:B) Show the region bounded by the curves y = x^2, x = 1, y = 0, generates 3 times as much volume when revolved around the y-axis as when revolved around the line x = 1.

IV. Either:A) compute the arclength of the portion of the curve x^(2/3) + y^(2/3) = 1, lying in the first quadrant;
Or:B) compute the surface area of the paraboloid formed by revolving around the y axis, that portion of the parabola y = (1/2)x^2 lying between x = 0 and x = sqrt.(24).

V.a. A tank shaped like an inverted right circular cone (vertex down), with base radius = 4 feet, and height = 8 feet, is partially filled with a liquid up to a height of 4 feet from the vertex. If the liquid weighs 1 pound per cubic foot, use calculus to compute the work required to pump all the liquid to the top of the tank. (Remember to calculate the work, in “foot/pounds” for one infinitesimal slice of volume and then integrate.)
b. Archimedes knew the center of gravity of a cone is 3 times as far from its vertex as from its base, and the volume is 1/3 its base area times its height. Use this information to recalculate the work done in part a, without calculus. [Do your answers agree?]

VI. Use standard techniques (parts, substitution, partial fractions, not power series) to find elementary antiderivatives:
a. ∫e^x cos(x)dx b. ∫cot(3x)dx c. ∫arctan(x)dx d. ∫(x^4)dx/(x^2 - 1) e. ∫dx/(sqrt(1+x^2).

VII.a. Use power series to solve the differential equation y’ = 2xy, y(0)= 1.
I.e. assume y = a(0) + a(1)X + a(2)X^2 + a(3)X^3 + a(4)X^4+…..
Compute y’ =
Compute 2xy =
Set equal the coefficients of like powers of X, in the series for y’ and 2xy, and express all the coefficients in terms of a(0).
Using y(0) = 1, find the coefficients a(0), a(1), a(2), a(3), a(4),…. And write out explicitly the corresponding part of the series for y.

b. Now use the separation of variables technique to solve the equation y’ = 2xy, for a familiar function y. Does it appear to be the same solution found in part a.?

VIII. Let f(t) = (1+t)^r.
a. Compute the following values of f and its derivatives at t = 0:
f(0), f’(0), f’’(0), f’’’(0), …..f^(n)(0).
b. Write down the first 4 terms of the Taylor series for f(t) centered at t = 0.
c. Simplify the fraction |a(n+1)/a(n)|, compute its limit, and find the radius of convergence of the Taylor series in part b.
d. Setting t = [-x^2], r = -1/2, give the first 4 terms of the Taylor series expansion of g(x) = 1/sqrt(1-x^2).
e. Integrate the series from part d. to find the Taylor series for arcsin(x).
f. Set x = ½ in part e., multiply by 6, and use the first two non zero terms to give an ancient Egyptian approximation to pi. (The first term is the “biblical” approximation.)

IX.a. Consider the triangle with vertices P = (2,1), Q = (3,3), and R = (5,1). Show the “medians” of this triangle meet in a point 2/3 of the way along each median from the corresponding vertex as follows.
Find the vectors pointing from one vertex to another: (recall Y-X points from X to Y.)
(Q-P) = ? (R-P) = ? (R-Q) = ?

Find the midpoints of the sides by adding to one vertex, half the vector pointing from it to the other vertex:
X = Midpoint of PQ = P + (1/2)(Q-P), Y = Midpoint of PR = P + (1/2)(R-P), Z = Midpoint of QR = Q + (1/2)(R-Q).
Then add to each vertex 2/3 the vector pointing from that vertex to the opposite side:
I.e. P + (2/3)(Z-P) = ? Q + (2/3)(Y-Q) = ? R + (2/3)(X-R) = ?
[If these points are all equal, you are done, if not, something is wrong.]

b. Find the perpendicular projection from the point Q to the side PR as follows:
All points on PR have form S = P + t(R-P) for some 0 ≤ t ≤ 1. Find t such that the vectors (S-Q) and (R-P) are perpendicular.
c. What is the angle of the triangle at the vertex R?

 

[uman]

How long do the students have to complete that exam?

 

[mathwonk]

3 hours but there is also an hour break before the next exam and i never run anyone out until the next class comes in, so at least 3 and 3/4 hours. i also give generous help and answer almost any question someone may have. i consider the exam a teaching opportunity and am willing to remind people of things they have forgotten.

notice there are several questions in which one is asked to do the same problem two different ways, and to compare answers. you do not want to be one of the people (they do exist) who simply say, no my answers do not agree, and yet do not go back and find the reasons and fix it.

if you expect to become a mathematician, that behavior is a "tell" that you are not in the right field.

 

[mathwonk]

euclidean geometry final

Try this, if you think you know high school (i.e. euclidean and archimedean) geometry (use limits, but no calculus):

5200 Final exam (Work any 6 of !-!X)
I. Congruence
Prove one of the SAS, SSS, ASA or AAS congruence theorems in neutral geometry.

From now on assume we are in a Euclidean geometry, i.e. assume EPP.
II. Plane areas
i) State the axioms for a Euclidean area function.
ii) Tell how to define an area function satisfying them.
iii) Say what technical facts need to be proved to justify the definition.
iv) Assuming an area function exists satisfying your axioms, argue that the area of a circle should be (1/2)Cr, where C is the circumference and r is the radius.

III. Similarity Assume the basic similarity result, that two triangles have the same angles if and only if they have proportional sides.
i) Prove for a triangle, the product “base times height”, is independent of choice of base.
ii) Prove triangles are congruent if and only if they are similar and have the same area.

IV. Pythagoras
Prove some version of Pythagoras, assuming whatever you need about similarity, area, or congruent dissections.

V. Concurrence
Which of the 4 basic triangle concurrence theorems are true without assuming the Euclidean parallel postulate, and which ones require it? Prove one concurrence theorem using whatever lemmas you need.

VI. Circles:
i) Show that two angles with vertices on a given circle are equal if they subtend the same circular arc. Is it true if they only subtend equal arcs? Why or why not?
ii) If a secant L joining 2 points on a circle meets another such secant M at a point p inside the circle, and point p separates L into segments of lengths a, b, and M into segments of lengths x, y. Prove that ab = xy. (In particular, if L bisects M perpendicularly, then ab = x^2.)


VII. Possible Constructions:
i) Given a pair of points one unit apart, and any other segment of length x, show how to construct a segment of length sqrt(x), and explain why your construction works.
ii) ) Say how to construct a regular pentagon in a circle of unit radius, and explain why your construction is correct. You may assume the regular decagon has edge length (1/2)(sqrt(5)-1).


VIII. Impossible constructions
i) Argue, in as much detail as you can, that a regular polygon of n sides can be constructed inside (inscribed in) a circle of radius 1, if and only if its edge length can be written in terms of integers using only the operations +, - , * , / , sqrt.
ii) Assuming that expressing the edge length of a regular polygon of p sides where p is prime, involves qth roots for every prime q dividing p-1, explain why 3, 5, 17, 257 are the only prime numbers p < 1000, such that one can construct a regular polygon of p sides.

IX. Surface area and Volume:
i) State the principles of parallel slices and magnification, and use them to explain why the volume of a pyramid should be (1/3) the product of its base area and height.
ii) Explain why the volume of a sphere should be (1/3)Sr where S is its surface area and r is its radius. (Hence if one knows the volume of a sphere one can obtain its surface area, and vice versa.)

X. Extra problems:
A.i) Compute the volume of a sphere, by showing that if a hemisphere and an inverted cone are inscribed in the same cylinder, then vol(cylinder) = vol(hemisphere) + vol(cone), using the principle of parallel slices. Hence the sphere has 2/3 the volume of the cylinder. (I handed out pictures of these.)
ii) Compute the volume of a spherical segment cut from a sphere by a plane, which has distance y from the center of the sphere.

B.i) Compute the surface area of a sphere, by arguing that it equals the lateral area of a cylinder circumscribed about the sphere, hence equals 2/3 the total area of the cylinder.
ii) Show the curved area of a spherical segment cut from a sphere by a plane, is equal to the area of a circle whose radius equals the line from the center of the segment (on the sphere) to the circumference of the segment. (In particular, the surface area of a hemisphere is that of a circle whose radius is a line from a pole to the equator, and the area of the sphere itself equals that of a circle whose radius is a diameter of the sphere.)

C.i) Prove that if two cylinders are inscribed in the same cube, but with their bases in different pairs of opposite faces of the cube, then the volume of the “bicylinder” (intersection of the two cylinders) equals 2/3 the volume of the cube. (Note that a plane parallel to the faces not containing bases of cylinders cuts each cylinder in a rectangle, and the bicylinder in a square. Hence replace the cone in problem A.i) by a square based pyramid. Presumably this is essentially Archimedes’ still lost solution.)
ii) It seems the surface area of the bicylinder is also 2/3 the surface area of the cube. Can you adapt the argument for B.i) to show this?
iii) Can you also find the volume and surface area of the tricylinder, using the same ideas used here for the sphere and bicylinder? (See Dr. Shifrin’s AMA talk on his website, for the volume of the tricylinder.)

 

[quasar987]

My current algebra teacher is an algebraic geometer. Maybe he went to the conference. He name is Abraham Broer.