Epsilon math camp

[Bostonian]

There is an Epsilon math camp for children ages 8 to 11 who are highly gifted at math. The rationale is described at http://www.epsiloncamp.org/why_epsilon.php and copied below. Many parents cannot afford such a camp (both the fees and the cost of being away with the child for 2-3 weeks). I wonder what books would be good for teaching young students gifted in math. Geometry is mentioned in the bolded section below. What would be suitable geometry books?

 

<beginning of excerpt>

 

The acute minds of the few dozen mathematically profoundly gifted students in the nation demand intervention and direct instruction. The MathPath summer program takes these kids provided they are at least 11 years of age during camp. Presently, the math summer camps for those under 11 years are not designed for the Epsilon-gifted. The new Espilon camp is an enrichment camp for students who are at least 8 years but under 11 years while at camp.

As at the MathPath camp, Epsilon gives the early experience appropriate for a future mathematician. If they choose a different profession, this early experience is likely to help than hinder. In fact, the role of the camp is to provide the setting for these students to grow in both social and academic domains. We are guided in this by the universal phenomenon of the musical banter between a mother and infant, enabling the infant to grow emotionally as well as to acquire the mother tongue. As the banter is not scripted prescription, so the topics covered at Epsilon are not the ones in the designed accelerated curricula. The topics, however, are to be mathematically relevant enrichment via interactive instruction where the mother is replaced by the mathematician, the theoretical physicist or the theoretical computer scientist.

 

A question arises. If MathPath caters to this kind of student, why can not the child wait till he/she is old enough to go to that camp? (The writer is the founder of MathPath.) The answer is that there are many issues that need to be addressed even earlier than at the MathPath minimum age. One issue is the need to do a critical study of Euclidean Geometry - a confident familiarity with the Elements as well as its shortcomings (proof-wise). This is pure mathematics accessible to the Epsilon-gifted at this tender age.

And we must show them what a certain property is really saying. For instance, the most popular theorem, the Pythagorean Theorem is more than what it appears. It is a statement concerning similarity. In fact, it is more than a statement about similarity. It is a statement of self-similarity under scaling, a property not shared by other Riemannian metrics of constant curvature! This is too much for the youngster of the 21st century, but at least we can take them up to self-similarity under scaling.

Further, at every opportunity these students must be shown the wider context - the general as it applies to the particular! Just as it is the instinct of a mathematician to inquire if a property or a statement holds in a more general context, it is a goal of instruction at the camp to show the more general context! Some properties hold in a general context, some do not! Both aspects are useful, particularly the latter for understanding. Granted, the instructor must take the student only to that generality that the student at the tender age at this camp can grasp.

Professor Robert Hunt, Deputy Director of the Isaac Newton Institute for Mathematical Sciences and a Fellow of Christ's College, University of Cambridge, commenting on Samantha Imafidon, a 9-year old math prodigy, stated: "Mathematics is a subject which some children latch on to very young - unlike, say, English Literature which requires a deeper emotional maturity - and they should always be encouraged to follow their interests, preferably through enrichment of the school curriculum."

 

Early intervention in mathematics is necessary for the Epsilon-gifted, for children become set in their ways - note how people grow up with the same religious belief that their parents have. The creative mind must be an inquiring mind which stands a better chance at pushing the boundary rather than accepting the boundary as the boundary. The way of mathematics - proceeding from answers to questions [1] - must become natural for the Epsilon-gifted at an early age. They must become comfortable early with the crisp language of mathematics that informs the reader with carefully chosen words and economy of expresson.

Almost all students up to grade 3 are interested in math; Epsilon Camp takes the Epsilon-gifted and endeavours with suitable instruction to blossom their early interest in to a passion. Of course, some students already arrive with the passion. But many do not, although they are interested. It is a tragedy not to find one's passion, but the greater tragedy is not pursuing one's passion.

 

The Epsilon-gifted need to meet their intellectual peers who are of their age. This is their most suitable social networking ... and they do not need to explain themselves to the peers!

 

Parenting the Epsilon-gifted has many challenges. The Epsilon camp also runs a parallel workshop on the same campus for the parents.

Edited May 18, 2011 by Bostonian

[MBM]

Kathy in Richmond is teaching there this summer, I believe. If she sees this, I'm sure she'll comment.

 

Btw, I think it sounds like a great idea.

[kokotg]

Math Olympiad starts in 4th grade. DH ran a group this past year, and the kids had a good time (we had one kid who was much younger than 4th grade by age, too, but really advanced at math; Math Olympiad was fine with that). So that's one idea for a way younger kids can get together to do math and hang out with other math-loving kids.

 

(and can I just pat my non-mathy self on the back for getting the "epsilon" reference? My FIL was friends with Erdos, and DH remembers being called an Epsilon when he would visit them).

[joannqn]

Honestly, I have no idea what that passage is really trying to say.

 

One of my big struggles with being a low income parent is the cost of all things enrichment. I can't afford to send my kids to a regular summer camp let along a specialized camp for the gifted. It drives me batty. I know he would probably love these classes and camps yet they are so very far out of our reach.

[Kathy in Richmond]

 

There is an Epsilon math camp for children ages 8 to 11 who are highly gifted at math. The rationale is described at http://www.epsiloncamp.org/why_epsilon.php and copied below. Many parents cannot afford such a camp (both the fees and the cost of being away with the child for 2-3 weeks). I wonder what books would be good for teaching young students gifted in math. Geometry is mention in the bolded section below. What would be suitable geometry books?

 

One issue is the need to do a critical study of Euclidean Geometry - a confident familiarity with the Elements as well as its shortcomings (proof-wise). This is pure mathematics accessible to the Epsilon-gifted at this tender age. And we must show them what a certain property is really saying. For instance, the most popular theorem, the Pythagorean Theorem is more than what it appears. It is a statement concerning similarity. In fact, it is more than a statement about similarity. It is a statement of self-similarity under scaling, a property not shared by other Riemannian metrics of constant curvature! This is too much for the youngster of the 21st century, but at least we can take them up to self-similarity under scaling.

 

 

Hi,

 

Yes, I'm helping to run Epsilon camp's first summer. We do realize that costs and scheduling make attendance impossible for some families. While we can't do anything about parental availability to take two weeks off to attend camp with their kids, we are working toward having some financial aid in place before too long (not this year, though). Getting a non-profit camp off the ground is all we can do this first summer.:001_smile:

 

ETA - joannqn - Have you pursued Davidson for your child? I know that some of our families w/ Davidson Young Scholars are applying for funding through that institute. I'm so sorry!! I totally understand financial constaints (have been there & done that!). Honestly, I know that the tuition cost looks out-of-this-world, but our camp is making zero profit. It all goes toward the expenses of putting on a first-class camp...

 

As for introducing the proper concepts of Euclidean geometry to eg/pg youngsters of this young age, there is no wonderful textbook out there that I know about yet. Our faculty and Dr. Thomas (a wonderful geometry teacher by the way) are developing our curriculum right now. It extends over several mathematical areas, not just geometry. But the children will be introduced to Euclid's elements in class, and the presentation will be accompanied by lots of hands-on ruler and compass constructions.

 

I'm not sure how to replicate this experience at home. We're requiring that our students have already been exposed to basic geometry at home or school, including familiarity with similar and congruent triangles. So assuming that the basics are already in place, the best I can do to give you a flavor of Epsilon's approach would be to try something like Robin Hartshorne's Geometry: Euclid and Beyond along with a copy of Euclid's Elements. Hartshorne is on the faculty at MathPath and is a fantastic geometry teacher there; I'm making the assumption that his book is written equally well (I only have my dd's copy in front of me which I've perused briefly; I've not used it directly myself). He does treat the Pythagorean theorem in the way suggested in the excerpt you quoted.

 

You'd definitely have to have the time to read this book to/with your child slowly. It's not something to just hand over to a child of Epsilon camp age. It would take an enthusiastic partner working with the child every step of the way. And you'd have to digest it in little bites at a time. But looking through it, I think that the beginning of the book could work with a motivated parent & child pair.

Edited May 17, 2011 by Kathy in Richmond

[Bostonian]

Thanks to Kathy for her informative reply. I will look at the Hartshorne geometry book. It gets good reviews on Amazon, but I suspect its primary audience is undergraduate math majors. The AOPS algebra math book I bought looks approachable to me, and I will probably get the AOPS geometry book.

Edited May 17, 2011 by Bostonian

[joannqn]

ETA - joannqn - Have you pursued Davidson for your child? I know that some of our families w/ Davidson Young Scholars are applying for funding through that institute. I'm so sorry!! I totally understand financial constaints (have been there & done that!). Honestly, I know that the tuition cost looks out-of-this-world, but our camp is making zero profit. It all goes toward the expenses of putting on a first-class camp...

 

No, I haven't. Like I said, I wouldn't even know how to go about doing that. I don't have test scores. I can't afford to do testing. I don't know what to put into a portfolio or if we'd have what is needed for a portfolio. And I don't know if it is worth it.

 

We have done SCAT testing for John Hopkins because we qualify for the fee wavers (it only cost us $20), and he qualified for their awards ceremony even though he tested at his grade level instead of age level and the proctor forgot to give him pencil and paper for the test. He did all of the math in his head. However, beside the nice paper certificate, it hasn't done us any good. I hesitate to go through the work of trying to get into Davidson if all we'll get from it is a list of classes and camps we can't afford to go to anyway.

[Kathy in Richmond]

Thanks to Kathy for her informative reply. I will look at the Hartshorne geometry book. It gets good reviews on Amazon, but I suspect its primary audience is undergraduate math majors. The AOPS algebra math book I bought looks approachable to me, and I will probably get the AOPS geometry book.

 

Yes, the AoPS materials are wonderful! You can't go wrong there, and they have terrific solutions manuals, too. Their geometry is challenging, but in a problem-solving way instead of a axiom-postulate-theorem sort of way, if that makes sense.:tongue_smilie: We got tons of mileage out of AoPS.:)

[Kathy in Richmond]

No, I haven't. Like I said, I wouldn't even know how to go about doing that. I don't have test scores. I can't afford to do testing. I don't know what to put into a portfolio or if we'd have what is needed for a portfolio. And I don't know if it is worth it.

 

We have done SCAT testing for John Hopkins because we qualify for the fee wavers (it only cost us $20), and he qualified for their awards ceremony even though he tested at his grade level instead of age level and the proctor forgot to give him pencil and paper for the test. He did all of the math in his head. However, beside the nice paper certificate, it hasn't done us any good. I hesitate to go through the work of trying to get into Davidson if all we'll get from it is a list of classes and camps we can't afford to go to anyway.

 

Yes, Davidson does have some financial aid for its Young Scholars. It's not automatic - you have to apply. Right now I have my fingers crossed that they'll come through for a few special youngsters that can't afford Epsilon camp otherwise.

 

You know, I didn't pursue Davidson for my family, though I'm sure that my kids would have qualified. I didn't see the point at the time. Like you, IQ testing was way too costly for us, and our homeschooling was going well without it. I printed off all their application stuff at one point, but it just sat on my desk for eons. I guess I didn't want to be bothered putting a portfolio together.:tongue_smilie:

 

Now that I'm working with Epsilon, I'm seeing Davidson from another point of view, and I kind of regret my decision. The networking with other families would have helped me as much as my kids. And I didn't know that they had financial help available for stuff like summer camps and courses. So, knowing what I know now, I would encourage you to look into it for your child. It couldn't hurt to try.:)

[Tattarrattat]

Thanks for the information Kathy! Very helpful!

Edited May 19, 2011 by Tattarrattat

[mathwonk]

" IQ testing was way too costly for us, and our homeschooling was going well without it. "

 

I am a new member and one of last summer's geometry professors from epsilon camp. Just a comment on testing cost. When our kids were young, we had them tested for free at the University of Georgia gifted center, where Paul Torrance was director. Maybe they needed subjects and made a special exception, but it worked for us.

 

I especially enjoyed meeting Paul Torrance. He was very inspiring and encouraging. His books on giftedness helped us as well.

[onaclairadeluna]

I used to wish I had the support of Davidson's. It is hard to figure out what the heck to do with an extremely bright child. I finally gave up I essentially decided that *I* needed to be the expert. Money would be so nice. Many, many resources are free or affordable. It's easier than you think.

 

AoPS is really affordable all you really need are the books and Alcumus is Free.

MIT open courseware (and other open courseware, yale for example) all free.

 

You can get many teaching company tapes free from the library. I also buy one or two of these for Christmas each year. Santa it turns out is an intellectual.

 

Websites for general gifted support.

hoagiesgifted.org

giftedhomeschoolers.org

senggifted.org

 

Mailing lists like tag-max tag-fam tag-pdq and gifted homeschoolers are places where you can chat.

 

Also once your child is 11 they can apply to Mathpath.They offer generous financial aid.

 

Oh and of course this place has an amazing wealth of resources.

 

If you think your child might qualify for D and you have access to a good tester that you can afford, go for it. But if circumstances don't allow this it is not the end of the world.

[mathwonk]

Here is one that costs $1000 for 8 weeks of online camp, but they said they had lots of scholarship money available this summer.

 

http://euclidlab.org/programs/camp-euclid/program-description

[mathwonk]

I am just learning to post and this my second try at this remark.. We were part of a program where a student was learning to give IQ tests, so for us it was free. This was at the UGA Torrance gifted education center in Athens GA, long ago.

[serendipitous journey]

... for others who are following the geometry part of this thread. Found the following at Berkeley. If it doesn't read easily to you, please know it is the least opaque section of that document. I think :D.

 

"The Pythagorean Theorem, also known as Euclid I.47 (i.e., Proposition 47 in Book I of the Elements), says that the areas of the squared built on the catheti [the shorter sides, adjoining the right angle] of a right triangle add up to the area of the square built on the hypotenuse: A + B = C. It turns out that Book VI of the Elements contains a generalization of the Pythagorean theorem that seems much less famous. Namely, Euclid VI.31 asserts that A + B = C for the areas A, B, and C of similar figures of any shape built on the sides of a right triangle. The Pythagorean theorem is clearly the special case where the shape is square. ... it is not hard to figure out that the generalization, in turn, can be derived from the Pythagorean theorem since areas of similar figures scale as squares of their linear sizes."

 

basically, that says that if you imagine a right angle with each of its three sides used to form three attached squares (like figure 2 here), and you slide the size of the triangle up and down, the way the areas of the squares naturally slide up & down too remains mathematically true to the Pythagorean theorem. Not only that, you could attach ANY SHAPE to each of the three sides, and as long as all three versions were similar (true scales of each other), their areas will show this same relationship. Neat-o tostito, no?

 

[this next chunk seems germane and I liked it, but don't know squat about Riemann whoseits so can't be sure ... you might like the definition of similarity, which the Epsilon camp finds so important a concept]

 

"... the true nature of the Pythagorean theorem as a statement based on similarity is revealed. What is "similarity" after all? From the abstract point of view it is conformal isometry: an isometry of a metric space with itself equipped with a rescaled metric. This explains why the Pythagorean theorem is a genuinely Euclidean phenomenon (and not only in the historical sense of the word): among all Riemannian metrics of constant curvature only the Euclidean one admits nontrivial conformal isometries. Perhaps the scaling self-similarity property of the Euclidean geometry is that fundamental balsamic ingredient that makes the Pythagorean theorem ageless."

 

... posted just in case anybody else was curious ...

Edited June 8, 2012 by serendipitous journey

[quark]

nm! :)

Edited June 9, 2012 by quark

[quark]

I am just learning to post and this my second try at this remark.. We were part of a program where a student was learning to give IQ tests, so for us it was free. This was at the UGA Torrance gifted education center in Athens GA, long ago.

 

Yes, we tested through a study at a local university. It's often so much more affordable (sometimes, even free). The disadvantage may be that you won't get a full interpretation of scores but you may be able to find a local licensed psych who will interpret the results over the phone for a much lower than normal fee.

[mathwonk]

I like the beautiful similarity derivation of a very general Pythagorean theorem. I want to point out however a certain circularity in reasoning involved here. I.e. in Book VI Prop. 2 of Euclid, the fundamental similarity result is proved using the theory of area (prop. I.39), the same theory of area that immediately implies the original version of Pythagoras (prop. I.47). One can also derive the theory of area from the principle of similarity, as is done in the AOPS geometry book.

 

Thus the theories of area and of similarity are essentially equivalent, but you have to start somewhere. If you do as Euclid did and start with area, then you can derive the original version of Pythagoras and then similarity, and then re-derive a stronger form of Pythagoras. But it is somewhat circular to claim that Pythagoras follows just from similarity if similarity is developed using area, as in AOPS. Thus the AOPS derivation is not logically correct.

 

Hilbert did show how to derive similarity without area, and thus gave a coherent and non circular development of area and similarity. This theory is very well described in Hartshorne's beautiful book: Geometry, Euclid and beyond.

 

I apologize for the pickiness of this comment, but this is the kind of slip up that a professional mathematician notices and feels he must point out. Indeed gifted kids also appreciate this kind of thing, even if their teachers do not. So yes, Pythagoras can be derived easily from a version of the similarity principle, but to be honest one should admit that similarity is not so easy to develop. One must either actually assume a theory of area that would already be sufficient to prove Pythagoras to even start the theory of similarity, or else work much harder to derive similarity without area. So in my opinion the quote above is misleading.

 

Just one man's opinion. In fact I have made the same comment on a site where professional mathematicians have made the same claim about pythagoras and similarity. The point is that Euclid's proof of Pythagoras did not even assume a concept of number, much less a principle of scaling. Moreover it works in a much more general world where in finitely small lengths exist, unlike the similarity approach.

 

http://mathoverflow.net/questions/40337/ingenuity-in-mathematics

 

about 10-12 answers down the page.

 

For what it's worth, a competition approach to math wants to use every possible trick available to solve the problem, but a mathematician's approach wants to understand the logic behind every assumption.

 

By the way I also disagree in a sense with the comment that only Euclidean geometry admits non trivial scalings ("conformal isometries"). I.e. the difference is that in spherical geometry for example there are many spheres, one for each radius. thus when you try to scale a spherical triangle, it is possible, but the scaled triangle lives on a sphere of different radius. I.e. you have to scale the whole sphere! In Euclidean geometry the scaled figure lives on the same plane. This is a property of the curvature, which in the Euclidean case is zero.

Edited June 9, 2012 by mathwonk

[serendipitous journey]

mathwonk, thank you! If I follow your argument, I'd have been fine if I'd only left out that second chunk, with its perspective of deriving the Pythagorean theorem from similarity & also its Euclidean imprecision: in the future, I shall be even more wary of things I can't follow!

 

I was sort of hoping the Euclidean scaling thing would relate to the treatment of parallel lines at infinity, which I guess is sort of related to curvature ... only because I've been reading a bit about infinity ... thank you for your explanation.

 

-- I don't suppose Hilbert's similarity/area work relates to Hilbert's transform? it is lazy to ask you, and I apologize {sheepish look} but if you know the answer offhand I would love to know -- it'll be a few days before I have time to look this up.

 

I just loved the scaling image, with figures of any shape. I had rather thought that to prove it with figures of any shape, not simply constrained by straight lines, infinitely small areas might come in [total admission that I have no real math intuition here, so please forgive if that's an awful blunder] but hadn't known that you would need infinitely small areas to deal with scaling. I suppose the smoothness of the scaling is the issue, scaling up and down in infinitesimally small increments; which seems obvious, once it's been pointed out (like so many things! :)) ... thank you for your thought-provoking response, for your gentle corrections, and for the links which will keep me busy when my time opens up a bit more!

[FairProspects]

.

[mathwonk]

from what i read on wikipedia, the hilbert transform is much more sophisticated, a tool in fourier series, and i don't know anything about it.

 

Actually the statements you linked above are arguably correct, I just point out that they can use some context. When I taught Euclid the first time I had trouble remembering all the tricky proofs in Book III that used area to derive facts about circles and triangles. When I saw how easy they were using similarity, I recommended my class to change the order of presentation and introduce similarity first, then use the easy derivations of the results like Euclid Prop III.35. We did the theory of similarity based on our prior knowledge of real numbers from calculus.

 

Last summer when I only had two weeks to teach this to kids who did not know calculus and real numbers, I was challenged as to how to get as far as similarity in only 12 lectures. I had been charged with presenting Books !-!V, and I knew similarity was in Book VI, but I learned that it was clearly a key property that was desired in the course. Stubbornly I declined to present it without theoretical justification.

 

Then I noticed that Prop. III.35 is also a statement of the basic triangle similarity theorem but in area form. I.e. the two secants form a pair of similar triangles in a a circle, and instead of saying their sides are in the same ratios A/B = C/D. it says equivalently that the rectangles they form have the same area, i.e. that AD = BC! Here multiplication of two line segments simply means the area of the rectangle they form.

 

Thus I was able to derive the basic principle of similarity, Prop. VI.2, in area formulation, from a much earlier one, Prop. III.35, using the Props in Book II to justify the algebraic properties of area. I was really proud of this, although I learned afterwards that of this had been noticed many decades before. The bonus was that we were able to simplify all the propositions of Books III-IV following this one, since we now had similarity of triangles available. (It is possible to see that every pair of similar triangles can be embedded in a circle as the ones are in Prop. III.35.) This simplified the proof of Euclid's, Prop. IV.10, for constructing a pentagon, by rendering the brilliant and tricky Prop. III.37 unnecessary.

 

This is all in my free epsilon camp notes on my webpage at UGA math dept, near the bottom of the page, #10.

 

http://www.math.uga.edu/~roy/

 

I think I have already done so, but I specifically recommend these great books which are free: Euclid's Elements;

 

http://aleph0.clarku.edu/~djoyce/java/elements/elements.html

 

(there is also a lovely paperbound edition from Green Lion for reasonable purchase.)

 

and Euler's Elements of Algebra. http://www26.us.archive.org/details/elementsalgebra00lagrgoog

Edited June 9, 2012 by mathwonk

[quark]

Not at all an expert so please pardon this question if it sounds silly or disconnected. Is it possible to demonstrate what you guys are discussing (Ana and mathwonk) using Geogebra? Any recommended materials to help visual learners create constructions in Geogebra so we can understand this discussion better?

[mathwonk]

i don't know the term geogebra but it seems coined to connote what is in euclid book II. there euclid proceeds as if the product of two segments means the rectangle they form sides of. Then the problem of when two rectangles have equal area is the same as when the product of two segments (think of their lengths) are equal.

 

He proves geometrically certain basic algebra results like A(B+C) = AB + AC. This just means visually that a rectangle with height A and base B+C has the same area as two rectangles both of height A, and one of base B and the other of base C. To prove it you just stick the last two rectangles together, or else you subdivide the base of the first rectangle.

 

He also proves stuff like (A+B)^2 = A^2 + 2AB + B^2, by looking at the four regions you naturally see in a square whose sides are both (A+B). I.e. there is an A square and a B square and two AB rectangles. I first saw this demonstration as a senior math major at Harvard from the famous psychologist of learning Jerome Bruner in 1965, and wondered why no one ever showed me before how easy that is.

 

These are propositions II.1 and II.4 in this version of euclid.

 

http://aleph0.clarku.edu/~djoyce/java/elements/bookII/bookII.html

 

 

Unfortunately!! there are no pictures in this version. That pretty much ruins its usefulness as a visual learning tool, i,e, as geogebra, unless you draw them yourself. let me hunt up a better version. (But he does give the equivalent symbolic algebra.)

 

Ohboy, here we go! it has pictures and even titles Book II as "geometric algebra"! are we getting there?

 

http://farside.ph.utexas.edu/euclid/Elements.pdf

 

 

On p.104 the circle has two secants forming two triangles (not fully drawn) with vertices ABE and DCE, and they have equal vertical angles, and also equal base angles (because those base angles cut off the same arcs of the circle). So the triangles have equal angles and we want to prove their sides are in the same ratios. But this means that AE/BE = DE/CE or equivalently , without defining what that means in terms of numbers, that rectangles (AE)(CE) = (BE)(DE), exactly the statement of Euclid's Prop. III.35.

 

Compare with Prop. VI.2 page 157, where the fact that the bases are parallel is equivalent to saying the base angles are equal.

Edited June 9, 2012 by mathwonk

[quark]

I know this sounds dramatic and all, but mathwonk, I am grateful that the stars aligned to bring you here. Thank you from the bottom of my heart!

[mathwonk]

thank you. I want to acknowledge after reading the berkeley link containing the remarks about similarity and pythagoras, that their author, Alexander Givental, is a MUCH better mathematician than I am. Hence everything he said is certainly literally correct. i still do not back down from anything I said however.

 

I conclude that he was interested in utilizing the power of similarity to make pythagoras seem more understandable, and was willing to overlook the task of justifying the concept of similarity rigorously. Sometimes ones goal is to unveil the insight behind something, even if it means assuming tacitly something which is intuitive, even if still hard to justify by logic.

 

I myself have also oscillated between using similarity to make area theorems easier and doing the opposite, using simple area concepts to define sophisticated similarity theorems.

 

there is another reason which I have suppressed, for doing area first. Namely there is a hypothesis called archimedes axiom, that says any finite segment can be repeated until it covers any other finite segment. this is needed to use the similarity approach of euclid, but not to use the area approach.

 

Thus the area first approach is more general and works even in non archimedean worlds, unlike the euclidean similarity approach. Professor Givental knows this of course, but his article was apparently serving a different educational purpose.

 

I recall a similar explanation of a problem solution by the great V. Arnol'd where he reduced a certain problem to the "obvious" fact that a continuous curve joining two opposite corners of a rectangle must meet a continuous curve joining the other two opposite corners. This makes the problem certainly believable. the problem is that obvious fact is rather hard to actually prove.

 

It is very helpful to the average person, or anyone, to see something clearly and simply from the right point of view. A few picky individuals still want to see the full proof with all the details.

 

(spoken by a 70 year old larvae.)

 

Ok, sorry, I realize this is mostly math jargon.

Edited June 9, 2012 by mathwonk

[mathwonk]

To try again to add something to the similarity remarks above, ones explanation always depends on ones goal. Professor Givental states that in his experience the similarity proof was the first one some of his friends had felt they really understood. So he was willing to use the concept of real numbers, which most people today find familiar, to illuminate the Pythagorean theorem by giving it an explanation. He wanted an argument that explained why the theorem is true rather than just a clever demonstration that still left us amazed at what we had seen demonstrated. he wanted his audience to say "oh THAT'S why that theorem is true." I now appreciate better his motivation.

 

However my goal is motivated by the fact that after teaching calculus for 40+ years to average college students, I have found that most in fact do not know anything useful at all about real numbers. Hence I was motivated instead to use the historical approach to Euclidean geometry to explain real numbers. I wanted my students to say "Oh THAT'S what a real number is, namely a way to compare lengths of line segments."

 

Of course I could achieve both goals by carefully explaining lengths and real numbers and similarity, as I did in my geometry class in college, and THEN doing Pythagoras as Prof Givental does, but that takes more time. Besides, the geometry is by definition more elementary than the similarity, since it takes less preparation. So in my opinion the Proposition VI.19 of Euclid cited by Prof. Givental, which establishes the fact that area scales as the square of length for similar figures, is more basic and hence more important than the relatively easy application made of it to reprove Pythagoras.

 

Moreover this principle is not at all clear to my incoming college students. In deed this is why area formulas tend to have squares in them π r^2, and so on, but kids actually do not know this. So he is taking something basic, perhaps intuitive, but difficult, for granted and using it to derive something else easily. So to me his easy argument is easy precisely because he left out the hard part! Nonetheless he may be pedagogically correct in this, since it seems to appeal to many people.

 

I think I have said enough to make it clear I believe there is no absolute best way to do anything, it depends on your point of view. As long as you have a pedagogical goal, you can feel good about trying to achieve it! But it does help if we know why we are doing something the way we do. Or as my brilliant colleague used to say, if you want to achieve your goal, first you have to have one.

 

This may also explain the endless discussions I have heard in coffee rooms (and online forums) over the "right" way to present something. The speakers simply have different goals in mind. If your goal is to keep your child engaged and thinking, then anything that achieves that should be enough, without worrying too much whether the curriculum they are using is approved by some official body or other. Just my view.

Edited June 11, 2012 by mathwonk

[mathwonk]

Let me pose a question in this regard about the AOPS treatment of area and similarity. In the early pages as I recall, area of a triangle is defined as (1/2) base times height. Then later the basic theorem on similarity of triangles, Euclid's Prop. VI.2, is proved using area, as Euclid does. What is odd about this? Well if you think about it, a triangle has three potential sides to be used as the base. Why does one get the same area from all three choices? I think I had never thought of this in my entire school career as a child. But in fact this proof requires the theory of similarity! So to use similarity to define area and then to use area to treat similarity is "circular" reasoning. I may have missed something in their explanation, but if not, then I wonder what goal did the AOPS authors have in mind for doing this?

[serendipitous journey]

I know this sounds dramatic and all, but mathwonk, I am grateful that the stars aligned to bring you here. Thank you from the bottom of my heart!

 

Me, too! Am reading furtively, and sporadically, while the boys Do Things to the yard and Button gets his summer education ... this is a wonderful conversation to be having. Am mulling over the AOPS question mathwonk posed, and also quark's call for visually-parseable math (and also all the other math/math-education above)...

[mathwonk]

Here's another fun observation about congruent triangles. I have seen it argued in books that SSS congruence for triangles is illustrated by the fact that a triangle made of three stiff straws with string running through them cannot be wobbled. I.e. there is no way to slightly alter this triangle without changing the length of the sides, even though the angles are not rigidly fixed. The argument is that the lengths of the three sides determines the angles.

 

But is this true? If you think about it all it says is that there is no other triangle "near" this one with the same sides. But so what? Does that mean no other triangle with the same sides could exist which is a discrete distance away from this one? I.e. just because every triangle which is a small variation of this one has different sides, why should EVERY other triangle have different sides?

 

Think of integer points on the number line. If you wiggle an integer a little, it is no longer an integer. E.g. there are no integers other than 1 in the small interval [.8, 1.2], but does that prove 1 is the only integer? Well, no.

 

 

It doesn't work for triangles either, remember that although SSS and SAS and AAS and ASA all imply congruence, that SSA does not. I.e. for a given pair of sides and an angle not between them, there are usually exactly two non congruent triangles with those same two sides and angle. But they are not too near each other.

 

I.e. imagine a triangle with two given vertical sides, and one given acute base angle. There are usually two triangles with that data depending on how you choose the other base angle, to be acute or obtuse. I.e. you look at the shorter vertical edge and let it flop over to point in the other direction. If it points left and makes say a 50 degree base angle , you flop it over until it points right and now makes a 130 degree base angle.

 

So there are two different triangles with the same SSA data, which cannot be slightly wiggled into each other, since they are a certain finite distance apart.

 

So if not being able to be wiggled, were enough to prove congruence, then SSA would also be a test for congruence, but it isn't.

 

 

So what goal do people have who give this illustration? Are they just happy to convey part of what is going on, or have they really confused what is called "rigidity" with uniqueness?

 

 

Gifted kids are much better at spotting these inconsistencies than the rest of us.

 

 

Here is a related question for you. What if two triangles satisfy side-side-obtuse angle? are they congruent? I.e. if two triangles have two sides equal and one obtuse angle equal, which is not necessarily between the two given sides, are they congruent?

Edited June 11, 2012 by mathwonk

[Ravi B]

Let me pose a question in this regard about the AOPS treatment of area and similarity. In the early pages as I recall, area of a triangle is defined as (1/2) base times height. Then later the basic theorem on similarity of triangles, Euclid's Prop. VI.2, is proved using area, as Euclid does. What is odd about this? Well if you think about it, a triangle has three potential sides to be used as the base. Why does one get the same area from all three choices? I think I had never thought of this in my entire school career as a child. But in fact this proof requires the theory of similarity! So to use similarity to define area and then to use area to treat similarity is "circular" reasoning. I may have missed something in their explanation, but if not, then I wonder what goal did the AOPS authors have in mind for doing this?

Which AoPS book did you read? I co-wrote their prealgebra book, but am not sure if that's the book you had in mind.

[mathwonk]

Hi Ravi, I think it was the geometry book. I learned a good bit from it, but I was puzzled by the area treatment I mentioned. All my parents were enthusiastic about the high quality of the AOPS series, so I may have tried to justify my existence by finding something to comment on. Is there a discussion of triangle area in your pre-algebra book? How do you do it? The geometry book was concerned with proofs as well as formulas, which is why this caught my eye. The pre algebra book may not be into proofs as much, I would guess, but I do not have one handy.

 

All my life I explained area of a parallelogram to my calc classes by reducing to that of a rectangle by subdividing and reassembling, but without noticing that the parallelogram I drew always had one upper vertex lying over the base, so that one subdivision suffices. It had never dawned on me to allow also a more skew parallelogram as Euclid does. Some of my 9 year old girls immediately (and I more slowly) saw how to reduce one case to the other by repeating the argument, but that uses archimedes' axiom, which euclid's argument does not.

 

After settling a parallelogram we always then treated a triangle as half a parallelogram. But again it never dawned on me to discuss what happens by choosing a different base for the triangle, i.e. independently proving the two different parallelograms and the two resulting rectangles do have the same area.

 

I enjoyed your post on cyclic functions by the way.

Edited June 12, 2012 by mathwonk

[wapiti]

Is there a discussion of triangle area in your pre-algebra book? How do you do it? The geometry book was concerned with proofs as well as formulas, which is why this caught my eye. The pre algebra book may not be into proofs as much, I would guess, but I do not have one handy.

 

You can get some idea from the AoPS Prealgebra videos, if you have time to sit through a few. I'd start with this one:

 

Section 11.2: Areas of Rectangles and Right Triangles

[mathwonk]

Thank you. I watched enough to see that this is a very intuitive discussion, with no proofs to speak of. Thee videos assume the existence of real numbers, and use them to define areas of triangles, by dropping perpendiculars to subdivide the triangle into two right triangles. No mention is made of the fact that there are three ways to drop these perpendiculars, and hence three candidates for the area. There is no mention of the need to show these three numbers are the same. So there is no attempt to make this discussion mathematically complete, or even to discuss what would be needed.

 

By comparison, the epsilon camp notes on my website are on a college level (more precisely, they were written for 9 year olds with the intelligence of college students), even though the class was 8-10 years old, and these videos are on an elementary school level. Still, these are the way to begin. My epsilon class had already seen this sort of thing and more, and I was charged with taking them higher. At least one of the epsilon camp 10 year olds had already sat through calculus classes. Others worked through some of Euler's algebra book on solving cubics by email.

 

I believe in exposing all kids to the deepest facts visible in their subject. They don't have to have any special scores or classes under their belt to be magnetized by interesting questions. You never know in advance which one will have a useful insight.

 

This is maybe esoteric, but if one uses real numbers to measure lengths of sides of triangles, shouldn't a student be able to answer the question: "what is a real number?" or "how do you multiply two real numbers?" if not, it is really "smoke and mirrors", i.e. they are using words that have no meaning for them. this is why euclid himself does geometry first without real numbers, and then uses the geometry to introduce real numbers essentially as lengths approximable by rational numbers.

 

historically geometry was developed first and used to motivate real numbers. today we pretend real numbers makes sense to kids just because they have heard the words, and we use real numbers to develop geometry. that to me is nonsensical. but opinions differ, as patrick swayze said in road house.

Edited June 13, 2012 by mathwonk

[jennynd]

I know this sounds dramatic and all, but mathwonk, I am grateful that the stars aligned to bring you here. Thank you from the bottom of my heart!

 

:iagree::iagree::iagree:

[crazyforlatin]

:iagree::iagree::iagree:

 

Me three!

[mathwonk]

you make me feel so welcome i am encouraged to say more. What if a kid sees a presentation telling her that the area of a triangle, i.e. the number of square feet that fit inside, equals half the product of the base times the height, i.e. half of the number of feet in the base times the number of feet that fit in the height?

 

So she goes outside and tries it, but it doesn't work. I.e. even if she makes the most precise measurement possible it never comes out right. What does that mean? Well the first reality is the margin of error in real life. But in fact our earthly world is round, so those formulas are all wrong. In fact there are no rectangles, and no parallelograms, on a curved surface like the earth. So in what sense are these geometry formulas true? Why do we teach them? Is there some flat world somewhere where they are true?

 

I think one plausible answer is that our world, although round, is roughly approximated by a flat world, at least if we don't stray too far from home, and also the flat world geometry is easier, so it is a good place to start. But we should make clear what its limitations are and what its assumptions are.

 

Why are these questions never raised in school? Why do we brainwash the kids to memorize facts that are not even correct in their own world where they live? We should know the answers to these questions, or at least ask them. Are we doing it because we think it is good enough as a beginning, or are we doing it because we never thought of anything else ourselves but what we were told in our school?

 

We always want to emphasize asking why things should be believed, and under what circumstances can the simpler approximate facts be helpful. On the other hand, we are all doing the best we can. We teach what we know, and admit that more will come later. And we want it to be interesting and fun. And we need to be sensitive to where the students are at the time. Sometime just: "here, isn't this a beautiful shape?" is enough. But sometimes more is possible.

 

By the way, how well known is it that there is an ancient Chinese text called the nine chapters, and commentaries by Liu Hui, which closely parallel some of the content of Euclid, but in an apparently independent presentation?

Edited June 13, 2012 by mathwonk

[serendipitous journey]

you make me feel so welcome i am encouraged to say more. What if a kid sees a presentation telling her that the area of a triangle, i.e. the number of square feet that fit inside, equals half the product of the base times the height, i.e. half of the number of feet in the base times the number of feet that fit in the height?

 

 

... it worked for us, but prob. because we worked from paper constructions: cut a rectangle, inscribe the triangle, cut out the extra pieces and fit them together so they equal the original triangle. Though I suppose your point fits here too: what with Imperfect Cutting, it never came out truly "square".

 

But why wouldn't this work on small areas? it also seemed to come out nicely on graph paper, esp. with right triangles inside a rectangle ...

[jennynd]

 

So she goes outside and tries it, but it doesn't work. I.e. even if she makes the most precise measurement possible it never comes out right. What does that mean? Well the first reality is the margin of error in real life.

I supposed this is just one way to make learning fun. The error probably will not come up in most case. U probably can get it close..

 

However, on pi day. I ask my son measure the diameter and circumference to calculate pi, well.. we never get 3.14 (he thought it was really sucks)... And that was a opportunity for me to mention uncertainty and get some statistic in his head. we did multiple measurement and start taking average. I even show him standard deviation. so original was meant to be a circle activity turned out to be a statistic day.

 

 

Why are these questions never raised in school? Why do we brainwash the kids to memorize facts that are not even correct in their own world where they live? We should know the answers to these questions, or at least ask them. Are we doing it because we think it is good enough as a beginning, or are we doing it because we never thought of anything else ourselves but what we were told in our school?

there is a simple answer to me. Teachers in this country.. hmm How do I say it politely... not that good.... I have not yet meet a teacher here I will say she is great. Their knowledge is limited and many of them, they themselves do not like math. let alone think math in deep way. They probably pray not to have a student like you :D

[mathwonk]

this is unfortunately a problem. but i must say i had a good one. she taught me all she knew herself, and wrote away to top schools for more advanced materials for us to use. then as a senior she stepped in and stopped me from only aspiring to go to the local college and got me into harvard over the phone for a life changing experience. and your child has a good one too, he has you.:001_smile:

 

you make an excellent point about the inability to make precise verifications of theoretical geometry in our real world due to the relatively huge errors involved. In the little book by Euler: Introduction to the analysis of the infinite, he computes pi to over 100 places almost perfectly. The one error in his book even makes me suspect an editor reproduced it incorrectly.

 

His trick is an improvement on the fact that pi/4 = 1 - 1/3 + 1/5 - 1/7 + 1/9 .......This series can be used to approximate pi as closely as desired, except that it takes too long to be useful. Euler figured out how to tweak this series to make the work go faster.

 

But another way to put the question of why doesn't the computation work in our real world, is to ask what world does it work in ? I.e. presumably euclidean geometry only works in a perfectly flat ideal world. How do we recognize such a world? That is where the axioms of Euclid come in. He tries to tell us in his axioms how to recognize an ideal flat plane. It turns out he left a few of them out, and it is fun to see which ones need to be added. That was the premise of our epsilon camp course. Euclid's 5 axioms and our additional ones are in the epsilon camp notes.

Edited June 13, 2012 by mathwonk

[mathwonk]

if "close" is close enough, why do we object when a state legislature wants to legislate that pi = 3, as is implied in the hebrew bible (as almost happened in the US in the 19th century)? What if they had wanted to legislate that pi = 3.14159? would that have been close enough? What about 3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679821480865132823066470938446095505822317253594081284811174502841027019385211055596446229489549303820?

 

And why do we tell kids that there exist irrational numbers? no physical experiment can ever detect that. see if you can verify with paper and scissors and measuring sticks that the diagonal of a unit square is not measurable with the same units as the edge, no matter how small those units are.

 

and try to construct a perfect pentagon and make from it a perfect dodecahedron, with paper and cardboard and scissors and straightedge. It is really hard to even get close. If we never get it close, do we tell the child, well i guess that construction of euclid is wrong after all?

 

and what does it mean that gauss showed how to construct a regular 17 sided polygon? do we think he actually made one? and what do we make of the fact that his argument also shows that a regular polygon with 65,537 sides is "constructible"? If you try to make that one, wouldn't the weight of the paper and the curvature of the earth begin to make trouble? It turns out that if we look up the word "constructible" in an abstract algebra book, it will mean something about the fact that a certain field of abstract numbers can be generated by taking only square roots! What does this have to do with paper and pencil drawings? Even the word constructible has apparently changed its meaning.

 

Of course the real world constructions suggest the plausibility of the theoretical ones, and are fun, when they come out, but they are not the criterion for determining correctness of the theory. Experiment is the criterion for correctness only in physics, which is concerned with this world, but math is concerned with an imaginary world where certain assumptions are made, often without regard to whether they hold here and now or not. And if we seem to note that the constructions only work well for small figures, what do we say to out child who asks whether we should rely on them for planning a trip to Rome, or to a distant planet? I.e. if we fly over Iceland to get to Europe, where do we point our space ship to get to Pluto? What geometry should we use? How do we find out what geometry to use?

 

 

Take one of euclid's simplest axioms, i think it is #2, that a finite line segment can be extended arbitrarily to a line. What does this mean? Does it mean that in repeating the segment over and over (Laying off copies of a finite ruler) we never run into an obstacle that stops us from repeating the segment again? This would happen if the plane were just the top of our kitchen table, i.e. if it had an edge we could not go beyond.

 

Or does it mean that in repeating the segment over and over that we never come back over the same ground? Note that we can repeat segments over and over as long as we wish on a round earth, but eventually, after about 25,000 miles, we come back to where we started. So the earth sort of satisfies euclid's second postulate, but not in the sense some might mean it.

 

So is a round earth a good model for euclid's axioms or not? In fact a round earth also satisfies the famous 5th or "parallel" postulate of euclid. why? we know there are no parallels on a round sphere. But euclid's version of the parallel postulate is a condition for two lines not to be parallel, hence it always holds on the sphere. (This is not the parallel postulate I was taught in school.)

 

so where do parallels come from in euclidean geometry? well to see we have to read his proof that they exist. it turns out to follow from his "exterior angle theorem", a theorem that was not given in my high school class at all. namely in any triangle, any exterior angle is greater than either of the two remote interior angles.

 

to prove this euclid uses an argument which depends on axiom #2, and he uses it in the strong sense that you never return to the same point when repeating a finite line segment. so we see that for his geometry, the sphere is not a model. So here is an example of an argument that seems to work on an 8x11 piece of notebook paper, but fails on a sphere if you use big enough triangles. (Has your child ever asked, "gee, would that proof work on a triangle a million miles high?)

 

by the way, to me this is a learning environment, where people interact with challenges and questions, and counterexamples. just working ones way through a book, no matter how carefully written, unless it is written by a master like euclid or euler or gauss, maybe not even then, is in my opinion unlikely to give this experience. so thank you for your challenges to my overly optimistic assertions. just remember to respect the challenges of your equally gifted kids also. and i know you do.

Edited June 13, 2012 by mathwonk

[mathwonk]

To return to the original topic, this is what epsilon camp was like every day. Those kids shouted out counterexamples to my incorrect claims and made me examine my hypotheses and shore them up. ("These are all possible regular convex solids with triangular faces." "I don't think so! I see another one!") It was really fun.

[serendipitous journey]

ahhhh, I see we have different approaches to math. :) Godspeed!

Edited June 13, 2012 by serendipitous journey

[mathwonk]

I support and agree with everything I hear you are doing. I'm just trying to suggest what may come "next'. They grow quickly. :)

 

As I said earlier, I suggest every teacher formulate some goals, and then try to achieve those goals. Different goals will naturally lead to different approaches.

 

Examples: I want my child to progress at least as fast as the local private gifted school curriculum and achieve superior standardized test scores, say in the top 5%.

 

Or, I want my child to get into MIT.

 

Or, I want my child to begin to become acquainted with the hands on meaning of abstract concepts.

 

Or, I want my child to think math is fun.

 

Or, I want my child to practice using her visual imagination.

 

Or, I want my child to learn to make logical arguments from precise hypotheses.

 

Or, I want my child to become acquainted with great historical documents.

 

Or, I want my child to complete the Singapore math program, or the AOPS series.

 

Or, I want my child to participate in formulating her own goals.

 

Or,...?????? it's entirely up to you.

 

 

These may all lead to different presentations. But if they help progress towards ones goals for that child, (and perhaps if the child tolerates it well, or better enjoys it,) no one can say anything else should be done. I certainly wouldn't. :D

 

My own personal presentation always responds to the reaction of the students in the class as the presentation is going on. I try to convey that math is alive and vibrant, and subject to question and verification. (That is why I am reacting to comments here! it' so much fun, and so enlightening.) For that same reason I cannot confidently recommend a program for a child I have not met. I just toss out ideas. I hope our different approaches complement each other.

 

Godspeed as well!

Edited June 13, 2012 by mathwonk

[mathwonk]

by the way here are some books with very down to earth and intuitive approaches to geometry that i just love. Daina Taimina and her husband David Henderson have some books building intuition for geometry before doing any formal stuff, and they include spherical as well as saddle surface geometry. And Daina shows how to get intuition for negative curvature (saddle surface) geometry from crocheting. The idea is that just as flat geometry is like that of a piece of paper, and positive curvature surfaces resemble the surface of our earth, negative curvature surfaces also come up in everyday life as ruffles on skirts and flowers and coral reefs! then Daina figured out how to crochet them, by adding a stitch every so often. This is to me amazingly cool. I would love to see them at epsilon sometime. For now we have their books.

 

http://www.amazon.com/s/ref=sr_tc_2_0?rh=i%3Astripbooks%2Ck%3ADaina+Taimina&keywords=Daina+Taimina&ie=UTF8&qid=1339620973&sr=1-2-ent&field-contributor_id=B001IXPX0U

Edited June 13, 2012 by mathwonk

[Emerald Stoker]

nm

[quark]

That is so beautiful mathwonk! Thanks for recommending the book!

 

My son and I have enjoyed Curve Stitching (accessible to young children while they develop confidence with crocheting?).

 

ETA: Some free resources here (watch out for one or two links that don't work)

Edited June 13, 2012 by quark

[serendipitous journey]

Mathwonk, I'm just bowing out because I think any differences in perspective are too fundamental for a thread to effectively address them, and we of course agree on much. DH and I think Button might do well at Epsilon when he's old enough to go; maybe (if the creek don't rise &c) you and I can chat at a meal sometime!

 

If I keep reading this thread I'll never get dinner cooked, much less finish planning our summer and next year! so I'm bowing out, but of course you can PM me. Thank you for a fun, engaging, and stimulating conversation.

 

blessings,

ana

 

ETA: just want to add that I am breaking from this thread simply because I'm tired and a bit overwhelmed at home, and think that mathwonk and I have very different perspectives but agree on all the math itself. So any apparent argument would be misleading, distracting, and not contribute to the conversation which has been rich & rewarding. I deeply appreciate this thread, and am so glad it happened -- it has been marvelous to think a little math! thank you all ...

Edited June 14, 2012 by serendipitous journey
precision. what else? :)

[Kathy in Richmond]

Took a board break for a couple of weeks and came back to lots of new stuff!

 

Mathwonk, welcome!

 

Though I have to say I'm surprised to see you over here in my homeschooling neck of the woods.:001_smile: I admit that I had a head scratching moment where I had to check to see which website I was on. Lol. It reminded me of when my eldest was in preschool, and we ran into his teacher at the KMart one weekend. He stopped dead in his tracks, got flustered, and whispered, "but Miss Valerie, she's supposed to live at school!" Grin.

 

Mathwonk did a superb job leading the Epsilon kiddos in geometry explorations last summer. If you take a peek at the main page of the camp website, you'll see him in action with one of our favorite little campers. Fond memories!

Edited June 14, 2012 by Kathy in Richmond

[mathwonk]

Hi Kathy! Great to hear from you. I am also very interested in homeschooling, having tried to school my own at home after "real" school, and supplement it in various ways. Also as an "educator", this environment seems to be where the really innovative and individual process of teaching can most likely occur, unencumbered with bureaucracy.

 

By the way, if anyone is as reluctant as I was to consider actually teaching geometry based on the original Euclid (I was over 65 before I even read Euclid!), this essay by Hartshorne made a big impression on me.....

 

Well the pdf document exceeds this site's limits, but if you type

 

"teaching geometry according to euclid" into google, it should be the first hit.

[mathwonk]

By the way, as may be obvious, one of my faults is a propensity for exaggeration. When I said earlier that someone making measurements outside will find they disagree with euclid, i should have said, at least they may once the triangles get so big they have vertices on different continents!

 

So serendipitous journey was exactly right to point out that the results look very good for small figures. In fact this is the basis of differential calculus! I.e. differential calculus is the science of approximating curved objects by flat ones in the most accurate possible way, at least near a given point.

 

So at each point of a sphere, there is a flat plane, the "tangent plane", that is the best flat approximation to that sphere, as long as we do not go far from the given point. I.e. for small figures, geometry on the Euclidean tangent plane is a very good approximation to geometry on the sphere.

 

This is true of every "smooth" surface. At every point of a smooth surface, there is a best approximating flat plane tangent to that surface, on which the geometry approximates the geometry on the surface, at least in the nearby region of that given point.

 

The process of finding the best flat plane approximating a given smooth surface at a given point, is essentially "taking the derivative" at that point. I don't know if this is accessible or not, but someone who has had calculus knows that the "derivative", or "partial derivatives", of the expression X^2 + Y^2 +Z^2 is given by (2X,2Y,2Z). At the point (0,0,1) this gives (0,0,2).

 

Then the remarks made above translate into saying that the tangent plane approximating the sphere

 

X^2 + Y^2 +Z^2 = 1, at the point (0,0,1) is the plane with

 

equation 0.(X-0) + 0.(Y-0) + 2.(Z-1) = 0.

 

I.e. the best flat plane approximating the sphere of radius one, at the north pole (0,0,1), is the plane 2Z = 2, or

 

Z=1, the flat plane touching the sphere just at the point (0,0,1).

 

So serendipitous journey is right, euclidean flat geometry does give approximately the right answer locally in small regions, and not just for spherical surfaces, but for any curved surfaces.

 

And differential calculus is nothing but a computational technique for finding the best approximating flat Euclidean plane at each point of a curved non Euclidean surface!

 

This gives one argument for learning euclidean geometry. Locally in a small region, it is a good approximation to any geometry. Since it is also simpler, it is the best one to learn, at least at first.

Edited June 14, 2012 by mathwonk