[HN Gopher] A seventh-grader student found a beautiful proof to ...
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A seventh-grader student found a beautiful proof to Thales' Theorem
(2002)
Author : samdung
Score : 116 points
Date : 2024-05-30 05:09 UTC (17 hours ago)
(HTM) web link (www.cut-the-knot.org)
(TXT) w3m dump (www.cut-the-knot.org)
| kristopolous wrote:
| I'd really like to read it in the language of the kid.
|
| Great mathematicians tend to start young. I don't question it,
| I'm just curious how the student phrased it.
| tromp wrote:
| I don't think the student's writing is available anywhere, but
| here is what Paul Lockhart said about it in his "A
| Mathematician's Lament" [1]:
|
| > To be fair, I did paraphrase the proof considerably. The
| original was quite a bit more convoluted, and contained a lot
| of unnecessary verbiage (as well as spelling and grammatical
| errors). But I think I got the feeling of it across. And these
| defects were all to the good; they gave me something to do as a
| teacher. I was able to point out several stylistic and logical
| problems, and the student was then able to improve the
| argument. For instance, I wasn't completely happy with the bit
| about both diagonals being diameters-- I didn't think that was
| entirely obvious-- but that only meant there was more to think
| about and more understanding to be gained from the situation.
| And in fact the student was able to fill in this gap quite
| nicely: "Since the triangle got rotated halfway around the
| circle, the tip must end up exactly opposite from where it
| started. That's why the diagonal of the box is a diameter." So
| a great project and a beautiful piece of mathematics. I'm not
| sure who was more proud, the student or myself. This is exactly
| the kind of experience I want my students to have.
|
| [1] https://fermatslibrary.com/s/a-mathematicians-lament
| matt3210 wrote:
| People seem to forget a lot that child math prodigies are being
| trained by parents to be math geniuses instead of enjoying being
| a kid. I had a 12 year old in my freshman CS class and he was the
| most miserable unhappy kid I've ever seen in my entire life.
|
| Edit: Pretty good points in the replies. Probably a lot of latent
| jealousy on my side in this. I should have thought on this more
| before commenting.
| saagarjha wrote:
| You don't have to be miserable to come up with a proof like
| this.
| fardinahsan wrote:
| Conversely, you can't come up with a proof like this if you
| don't love math.
| croes wrote:
| Why not?
|
| It may be more likely but not impossible.
| Larrikin wrote:
| What's the take away you want from this comment? Advancement of
| humanity, feeling bad for missed childhood fulfillments, some
| mix? Did the child actually meaningfully contribute anything to
| your class? Were they a real person? Right now there's a lot
| we've "forgotten" about child geniuses that we should all know
| from our interactions from them ?
| seszett wrote:
| Just to provide another data point, I also had a "math genius"
| friend at school and he was a well-balanced, fun and reasonably
| happy person.
| knightoffaith wrote:
| Some more data - the math geniuses I knew were actually
| happier, more well-rounded, and fitter than the average
| person at my former school.
| junon wrote:
| There was a trope - track and math.
| blueflow wrote:
| As someone who was enrolled in university starting from 9th
| class - i agree that this is not what children should be made
| to do. Adults were projecting weird ideas of success onto us.
| admissionsguy wrote:
| I was an extremely miserable 12 year old in a shit environment.
| Would have preferred to be miserable in a university class.
| TrackerFF wrote:
| When I was tutoring HS kids (as a grad student myself), we also
| had afternoon practices for the kids that wanted to compete in
| the math olympiads.
|
| Truth be told, some of the kids _loved it_ - but those were the
| kids that were really passionate about math, for the sake of
| math.
|
| But you also had a bunch of kids that seemingly hated every
| second of it. So why did they do it?
|
| - Some had pushy parents that wanted them to excel in
| extracurricular activities. Think typical tiger parenting...
|
| - Some had big ambitions about certain schools, and felt that
| they had to compete on a national level in _something_ , in
| order to stand out in the selection/application process.
|
| Granted, this was over 15 years ago. A couple of years ago I
| checked up on some of the kids (LinkedIn), and the most
| passionate kids were either now math/physics Ph.Ds, or Ph.D
| track.
| bradrn wrote:
| A counterpoint: I was accelerated by three year levels, and it
| was the best thing that could have happened to me. Otherwise I
| would have been intensely bored in high school for six years
| straight, with all the problems that would inevitably have
| brought.
|
| Of course, this depends so much on the person in question.
| Acceleration can be great for some people and awful for others.
| But it's not about 'enjoying being a kid' -- that's something I
| hear a lot, but I enjoyed myself a lot more in the higher
| grades where I was actually learning material I found
| interesting.
| smeej wrote:
| Can confirm: My parents made me go through the grades one at
| a time, despite the principal's recommendation that I start
| in 1st grade, move to 2nd mid-semester, do 3rd grade in the
| spring, and do two a year after that. It _was_ intensely
| boring. I can see why they thought a bunch of teens and young
| adults wouldn 't want to be friends with a much younger
| child, but I've always wondered if they would have been more
| willing to overlook the age difference than the other young
| children were willing to overlook the fact that the _only_
| thing we had in common was our age.
|
| Ironically, I found it decent preparation for life, though,
| which seems to be occasional bursts of furious activity
| punctuated by long stretches of repetition, which could
| easily become boring. I ended up becoming very good at taking
| responsibility for making my own life interesting.
| bradrn wrote:
| > I ended up becoming very good at taking responsibility
| for making my own life interesting.
|
| Oh yes... I also spent considerable time teaching myself
| interesting topics. It's a good skill to have. But I was
| still bored a lot of the time, and being able to do the
| interesting stuff in school too was a much better
| experience for me.
| smeej wrote:
| Ultimately I've had to get to a point where I realize I'm
| grateful for the life I live now, and accelerating
| through school would have changed a lot of it, such that
| I probably wouldn't have met some of the people most
| important to me, so I'm grateful for it in the long run.
| But I don't know that I'll ever stop wondering, and it's
| not a comparative gratitude, because I'm sure my life
| could have become beautiful like this in other ways. It's
| more a _practice_ of radical acceptance with gratitude.
|
| I will say one of the things that always made me laugh
| was that at each milestone, like finishing elementary
| school and starting middle school, my parents would try
| to tell me _now_ it would be different, because I was at
| a higher level. I remember pointing out to them all the
| same doofuses from the previous years were moving up
| right along with me, and it wasn 't just that I was
| _ahead_ of them, but that I was _faster,_ so this didn 't
| seem to help. I never did understand why they thought
| that would work!
| WillAdams wrote:
| It is unfortunate that this is not a core principle of
| education.
|
| The best school system I ever attended rigorously divided
| classes between social and academic --- for social classes
| (gym, health, social studies, homeroom) one attended at one's
| grade level, while for academic classes, (English, other
| languages, science, math) one worked at one's grade level
| (but with a cap of 4 grade levels through 8th grade, so a 4th
| grader couldn't take high school classes) --- after 8th grade
| this cap was removed, and students could take any classes
| which they could qualify for academically. To facilitate
| this, some teachers were also accredited as faculty at a
| nearby college, and if necessary arrangements were made
| either for college professors to come to the school and
| teach, or students travelled to the college.
|
| Most students graduated with at least a little college
| credit, and many received a college degree along with their
| high school diploma --- until the Mississippi State Supreme
| Court decided that it was illegal because it conferred an
| unseemly advantage on some students with no corresponding
| compensation for students who were unable to avail themselves
| of it academically.
| bradrn wrote:
| > The best school system I ever attended rigorously divided
| classes between social and academic
|
| This is such a great idea! And it sounds like it works well
| in practise, too. I've often pondered the idea of
| separating classes by ability rather than age, and this is
| a really practical method of achieving it.
| WillAdams wrote:
| It worked quite well --- helped that it was a small,
| cohesive school system with all grades in the same
| building/complex.
|
| A similar idea is to group children by reading level
| which makes classes function more smoothly since students
| finish reading assignments in similar periods of time.
| smeej wrote:
| Ironically I think a version of this had to happen in the
| one-room schoolhouses of yore. If you're the only
| teacher, you can't afford to split everyone by age. You
| kind of have to roll people into whatever lessons they
| can handle and do your best.
|
| It was "innovative," in a way I think we're coming to
| regret, to suggest that all 8-year-olds (for example)
| must be at the same level across every subject, and none
| of them should ever be taught alongside a group of mostly
| 6- or 10-year-olds.
| DontchaKnowit wrote:
| This is brilliant. Would love to see more of this in the US
| education system.
| gaurav_v wrote:
| Which school was this, and when did you attend?
| WillAdams wrote:
| The public school system near Columbus Air Force Base and
| I attended for 3rd and 4th grades (taking 7th and 8th
| grade English and science courses) in the '74 and '75
| school years.
| smeej wrote:
| What did you do when you moved and your new school was
| expecting you to do 5th grade work?
| WillAdams wrote:
| For reading, worked through a box of SRA booklets in a
| couple of weeks time and then went to the library for the
| rest of the year --- mostly I was bored and coasted until
| my classes caught up, and even then, I was able to get
| decent grades w/ minimal effort.
|
| Of course this all came to a head when I was a junior and
| the school system couldn't find a teacher for Calculus
| for myself and the couple of other students who wanted it
| which pretty much killed my college prospects, so aced
| the ASVAB, DLPT, and EDPT and enlisted.
| mwigdahl wrote:
| Chief Justice Diana Moon Glampers presiding, no doubt...
| dhosek wrote:
| When I was in kindergarten, I was one of a handful of kids
| who were invited to come back to the school after lunch (this
| was in the days of half-day kindergarten being the norm) to
| learn math with the first graders. I got kicked out for being
| disruptive.
|
| My mom, annoyed at not having her afternoons free, gave me
| math learning materials to keep me busy. Since she wasn't a
| teacher, she didn't really know what she was doing and I
| ended up entering first grade reading at a fifth-grade level
| and doing math at a third-grade level.
|
| I really should have been skipped ahead at least a year, if
| not two. The first-grade teacher had me teach the gifted kids
| in the back of the room while she taught the rest of the
| class, the second-grade teacher didn't really know what to do
| with me and I was bored out of my skull as a result (first
| and second grade swapped between the two teachers at lunch
| time with the first-grade teacher focusing on reading and
| writing, the second-grade teacher focusing on math and
| science).
| smeej wrote:
| One of the last times my parents moved, they sent me this
| big box they had of papers from my childhood.
|
| One of the ones that fascinated me was dated from when I
| was in 2nd grade. It was a big printout with rows and
| columns. Some of them contained numbers, but most of them
| just said "PHS."
|
| It was an oddly folded poster-size thing, and once I
| finally got it open, I found out the numbers were grade
| levels, and "PHS" stood for "Post High School."
|
| I had been 8 when I sat for this test. My parents had
| staunchly refused to accelerate my education.
|
| And they continued to refuse for ten. more. years.
|
| I might recommend that parents not send their children
| _proof_ if they went out of their way to stymie them.
| euroderf wrote:
| > I was accelerated by three year levels
|
| And you avoided being bullied ? Nice!
| graemep wrote:
| The post may not make it clear that this was a student of Paul
| Lockhart's whose main aim is not make maths fun. This is a good
| teacher, not parental training.
|
| It was also similar to how I taught my home educated kids - do
| fun stuff. It worked very well. Both love maths (and hugely
| enjoyed studying in general). My older daughter is doing a
| degree apprenticeship in electronic and electrical
| engineering[1] and the younger one got a 9 in IGCSE maths[2]
| and plans to do A level maths and further maths[3]. The younger
| one hated maths while she was in school, which is common, and
| which is the problem Lockhart was trying to solve.
|
| This is not about pushing kids or making them prodigies, it is
| about making them enjoy what they do, which leads to higher
| achievement and and happiness.
|
| That said, not all prodigies are miserable. Ganesh Sittampalam
| who broke the UK record for youngest graduate taught himself
| for fun until he got into university!
|
| [1] UK term for degree paid for by employer done while working
| (and getting paid). Good for parental wallet!
|
| [2] UK exams sat at 16. 9 is a top grade.
|
| [3] Closest American thing is APs.
| graemep wrote:
| Cannot edit anymore, and spotted a rather bad typo. It should
| read "whose main aim is TO make maths fun"
| lostlogin wrote:
| Having just been caught in a situation where a 'does'
| versus 'doesn't' typo in an email lead to a painful
| misunderstanding, this made me laugh.
| aj7 wrote:
| No parent can "train" a kid to be a math genius. There is a
| wide divide between a "math genius" and having a high SAT score
| for instance.
| JadeNB wrote:
| > No parent can "train" a kid to be a math genius.
|
| While that's true, there's lots of parents who don't believe
| it, or don't care if the end result isn't actual mathematical
| genius as long as it's some sort of distinction. (I'm in the
| happy position where my parents encouraged me to pursue the
| paths that I enjoyed, which wound up with some acceleration,
| but my getting in the local newspaper meant that we were
| besieged by the kind of parents who do just want to push
| their kids towards some pre-determined "success.")
| lern_too_spel wrote:
| While I might quibble with the parenting technique because
| I believe an interest in useful topics can be cultivated,
| I'm not going to knock those parents for developing the
| ability to do hard work. The rest of the world is better
| off with these kids becoming competent professionals due to
| parental control than it would be if they became wastrels
| due to parental neglect or encouragement to pursue useless
| hobbies. In the US at least, the much larger problem is the
| latter.
| JadeNB wrote:
| > While I might quibble with the parenting technique
| because I believe an interest in useful topics can be
| cultivated, I'm not going to knock those parents for
| developing the ability to do hard work. The rest of the
| world is better off with these kids becoming competent
| professionals due to parental control than it would be if
| they became wastrels due to parental neglect or
| encouragement to pursue useless hobbies. In the US at
| least, the much larger problem is the latter.
|
| I think that's a very optimistic view of the outcome of
| this sort of thing, although it could be that I just
| didn't describe the parental behavior well. This wasn't
| parenting telling their kids "hard work is worth it!",
| but parents who decided on the specific honors that their
| child would achieve, and would permit no deviation from
| the path, as a result of which most of these kids either
| burned out before achieving their goal, or rebelled,
| usually in self-destructive ways, as soon as they were
| given a little freedom from parental control.
|
| Different parenting styles work for different people, but
| for me (as a child in the US in the '80s) I think that my
| parents did the best possible thing, which was to instill
| in me lessons about the value and importance of hard
| work, but to make it clear that, in the end, the decision
| about whether I would pursue those values was up to me.
| It probably helped that, despite the many other
| advantages I enjoyed, my family was insufficiently well
| off to represent a guaranteed financial fallback for me,
| so that I knew that, one way or the other, I'd have to
| make my way in the world--which made it particularly
| appealing to me to be able to find a way to earn my keep
| by working hard for something I loved.
| bmacho wrote:
| Very untrue.
| dhosek wrote:
| My son who is just finishingh fourth grade has been demanding
| harder math problems than he gets in school. He learned how to
| calculate square roots by hand from a YouTube video and enjoys
| doing these problems on his own, checking his answers against a
| calculator. When I pick him up from school, when we're walking
| home, he demands that I teach him algebra. I've taught him how
| to solve basic linear equations ( _ax_ + _b_ = _cx_ + _d_ ) and
| also I have him take a number and try to find all the sums and
| differences of pair-wise factors (I haven't told him _why_ but
| people who remember high school algebra will recognize this as
| a skill essential to factoring quadratics) We're going to learn
| Trachtenberg Speed Math over the summer. He also enjoys coding
| and I remember being astonished watching him--without guidance
| --figure out how to rewrite a program he'd done in Scratch
| using Python (although right now JavaScript has become his
| favorite language).
|
| I've not pushed him into any of this (I really would prefer
| that he not pursue programming as a career as I fear that we're
| heading to a dark ages of software development, but he can do
| what he likes).
|
| His twin sister, meanwhile, has been demanding that I teach her
| social studies so I've been doing my best to give her bits and
| pieces of history and government from my memory. She doesn't
| demand the math, but I've noticed that she is paying attention
| during the algebra on the walk lessons as she'll occasionally
| jump in with an answer herself.
|
| My purely selfish side wouldn't object to them skipping a grade
| as my ex-wife seems committed to them doing K-8 in private
| schools (even though our local public schools are quite good
| except at the junior high level, but 6-8 is hell at any school)
| and skipping a grade would save me a year's tuition expenses.
| jacobolus wrote:
| I recommend you find a copy of Lenchner's book _Creative
| Problem Solving in School Mathematics_ (2nd ed.). Lenchner
| ran math olympiads for late elementary school / middle
| school for decades, and does a good job of covering a wide
| variety of useful strategies for approaching nontrivial word
| problems (i.e. real math problems, not just arithmetic
| drills). But you don't have to be interested in contests per
| se to get a lot from the book. In my opinion that's going to
| be more pedagogically valuable and more fun than
| Trachtenberg. Here's an Internet Archive scan
| https://archive.org/details/creativeproblems0000lenc
|
| Your kid would also probably really enjoy _The Number Devil_
| , a breezy novel about a kid who gets trapped in his dreams
| with a devil who won't stop telling him about elementary
| number theory, using various dream-world props.
|
| If you want more problems after that, there are a lot of
| great puzzles in Kordemsky's _The Moscow Puzzles_
| https://archive.org/details/moscowpuzzles3590000kord_m9a0, or
| you could take a look at Fomin, Genkin, & Itenberg's
| _Mathematical Circles: Russian Experience_ which your kid
| could probably just about handle.
|
| There are also some great puzzles and games explained at the
| website of the Julia Robinson Math Festival, https://jrmf.org
| which can be played with fairly basic easily available
| materials.
| akho wrote:
| Proving that a parallelogram with equal diagonals is a rectangle
| is an exercise in itself; I'd prove it through Thales' theorem,
| myself...
|
| Also, _Lockhart 's Lament_ is from 2002, so this post probably
| needs a (2002). It is very unlikely that the proof was new. It
| was certainly new to the seventh-grader, and a great result at
| that.
| lloeki wrote:
| > Proving that a parallelogram with equal diagonals is a
| rectangle is an exercise in itself; I'd prove it through
| Thales' theorem, myself..
|
| IIRC you prove it via angles.
|
| a) The sum of angles of any triangle is pi.
|
| b) The sum of angles of any quadrilateral is 2*pi.
|
| c) Since you rotate the triangle, opposite angles end up adding
| together. Because of the above, a bit of reasoning around
| symmetry shows that when diagonals are equal they can only add
| to pi/2 and that the other ones can only be pi/2. Any other
| angle leads to a contradiction.
|
| I mentioned angle values but (again IIRC) this can all be
| proven with compass and ruler.
| vidarh wrote:
| As the article mentions, in this particular case it's
| simpler: You have a point on the triangle that is also on the
| circle. Now you rotate the triangle 180 degrees, with the
| point following the circle. The point inherently must end up
| exactly opposite the original point, and so the diagonal
| formed between the old and new point must be the diameter of
| the circle.
| vlovich123 wrote:
| Um... small correction. Sum of angles in triangle is 180 not
| pi (right-angle triangle is 90 + 45 + 45). Similarly for
| quadrilateral it's 360 not 180 (a canonical being rectangles
| which is 4 90 degree angles).
| akho wrote:
| What do you think pi is.
| vlovich123 wrote:
| Apparently HN doesn't support the facepalm emoji.
| akho wrote:
| "Standard" Thales proof (via two isosceles triangles) is
| simpler than this.
| jasomill wrote:
| In case (a) and (b) aren't immediately obvious:
|
| The net change in direction around a triangle, quadrilateral,
| or any simple closed curve is a full rotation, i.e., 2p.
|
| The sum of the interior angle and the signed change in
| direction around a vertex of a polygon is p (consider the
| case of a "vertex" with interior angle p).
|
| So for a polygon with interior angles th1, th2, ..., th[?],
|
| 2p = (p - th1) + (p - th2) + [?] + (p - th[?]),
|
| therefore
|
| th1 + th2 + [?] + th[?] = np - 2p = (n-2)p.
| jacobolus wrote:
| Proving that a parallelogram ABCD with equal diagonals is a
| rectangle is an exercise, but not too hard. You can make two
| different triangles ABC and ABD with the same parallelogram
| base, where one has second side BC and third side (a diagonal)
| AC, and the other has second side AD and third side (the other
| diagonal) BD. These triangles are SSS congruent, therefore must
| have congruent angles (Euclid I.8) ABC and BAD. This gives you
| a parallelogram with 4 congruent angles.
| muro wrote:
| Nice. You can alternatively prove the negative: suppose it's
| a parallelogram with non-right angles, assume angle at A is
| "sharper". Thus, you can create two right angle triangles:
| AB'C with the base extended beyond B (= B') where AC is one
| diagonal and A'BD where A' is "below" D and BD is the other
| diagonal. As AB'C is larger than A'BD (because A' is within
| AB and B' is outside while the height is the same) AC can't
| be the same length as BD.
|
| Thus, no parallelogram with non-right angles exists that has
| equal length diagonals.
| akho wrote:
| Sure. The overall proof of Thales is then about as
| complicated as the "standard" one with two isosceles
| triangles.
| jacobolus wrote:
| Breaking proofs of trickier theorems up into a few parts
| and hiding the details of each part behind another theorem
| whose proof is self-contained and easy to follow (even
| better if it's "obviously true") is the basic idea of
| mathematics. Trying to make the top-level proof very
| clearly demonstrate, without a lot of bookkeeping or head
| scratching, that the theorem must be true is a wonderful
| goal.
|
| In my opinion it's a lot more obvious that a parallelogram
| with 2 equal diagonals must be a rectangle than that an
| inscribed angle intersecting a circle at ends of a diameter
| must be a right angle.
| akho wrote:
| Whether this proof is complete depends on what was known
| before. Was the lemma introduced by a teacher? was it not
| actually proven, and the student missed the hole in their
| proof? these are different teachable moments.
|
| I think the two facts, by themselves, are roughly
| equivalent in difficulty. Moving from Thales to equal
| diagonals parallelogram requires drawing an extra circle;
| moving from parallelogram facts to Thales requires either
| the construction in the article, or continuing the
| missing radius to get the same rectangle.
| jacobolus wrote:
| I don't understand why you are invested in nitpicking the
| work of an anonymous 7th grader from decades ago.
|
| Clearly, filling in all of the details of any formal
| proof in Greek style is going to require a careful
| knowledge of the axioms in use and some list of
| previously proven theorems which are allowed to be used
| without re-proof. Depending on which theorems are at hand
| already, one or another proof might be shorter or longer
| or more or less obvious. When trying to construct a whole
| mathematical treatise, the best order for the theorems so
| that each proof depends only on previously proven
| theorems is a tricky choice involving some trade-offs
| between pedagogical goals and concision of individual
| proofs.
|
| But all of that is missing the point.
| akho wrote:
| In the spirit of the posted proof, noting axial symmetry
| would be neater.
|
| (But then you need to know that parallelogram diagonals split
| each other in half, ...)
| tmaly wrote:
| I noticed the link on the link to Paul's post was not working.
| akho wrote:
| Lockhart's Lament is pretty well known in maths education
| circles. Calling it an "online article" is weird.
| dang wrote:
| Changed from 2008 to 2002 above. Thanks!
| rocqua wrote:
| I don't think the note "Since the triangle got rotated halfway
| around the circle, the tip must end up exactly opposite from
| where it started. That's why the diagonal of the box is a
| diameter." is needed, nor do you need to show the diagonals of
| the parallelogram are equally long.
|
| Rather, it suffices to simply state, the only parellelogram that
| is inscribed in a circle is a rectangle.
|
| For the less rigorous, that statement is obviously true. For
| complete rigor, it is sufficient to argue that the center of the
| circle, combined with the vertices of the parallelogram forms an
| isoceles triangle. So the centre of the circle must lie on the
| bisector all edges of a parallelogram. But on a non-rectangle
| parellelogram the bisectors of opposite edges never intersect.
| lupire wrote:
| If your informal proof is to draw a picture and say "it's
| obviously true by lookikg at it", then you don't need a
| parallelogram at all. Just draw the diametric triangle and look
| at it.
| rocqua wrote:
| Hence the addition of a formal proof based on the bisectors
| of opposite edges of a parallelogram.
| forgotpwd16 wrote:
| To be honest, I prefer the _unattractive and inelegant_ proof
| given alongside that one in Lockhart 's essay.
| lupire wrote:
| Lockhart missed the point a bit.
|
| His student's "proof" is an illustration, not a proof. There's
| no way to know if its circular or simply unfounded, since it is
| purely an appeal to intuition. That's only a part of
| mathematics. The correct thing to do, mathematically, is to
| validate the intuition by formatting it as a proper proof,
| based on non-circular axioms and theorems. In the paper itself
| he admits that his student's work was incoherent and needed him
| to rewrite it.
|
| He's right that 2 column geometry proofs are ugly, and could be
| presented better. This has been known for centuries.
|
| https://www.c82.net/euclid/en/book3/#prop31
|
| For children and for starting out, the pictures are great. But
| for mathematics, pictures are extremely limiting. 2D and 3D are
| notgod models for N-Dimemsions. ("Spiky balls" , for example).
|
| Mathematics is far, far more powerful than human eyes. The
| amazing thing about geometry is that the whole thing works
| without pictures! A blind person can be a great geometer,
| because geometry is axiomatizable. Meanwhile, Euclid's
| Elements, while an incredible achievement in its day, is not
| well-founded, relying on unstated axioms.
| JadeNB wrote:
| > Lockhart missed the point a bit.
|
| I think it's fairer to say that his point may not have been
| what you expected it to be. As a teacher myself, trust me, if
| a student comes to you and says "I came up with a proof!" and
| you say "no, you see, what you have is an intuitive
| explanation that can possibly be turned into a proof," then
| all that will happen is that that student will not be
| interested any more in exploring, or at least will not be
| interested any more in sharing their explorations with you.
| At that point, you have both lost.
|
| Lockhart is well aware of the standard of mathematical proof,
| and knows that, all else aside, this theorem is not in want
| of proof. His focus is on the fact that, if we want
| mathematics to remain a live profession, then we must improve
| our pedagogy, and help to train students who enjoy and want
| to pursue mathematics--even if it means occasionally
| accepting less than maximally rigorous mathematics from a
| seventh grader.
| zyklu5 wrote:
| While I respect your point about rigour I should say one has
| to be careful not least because the way the subject is
| presented in textbooks are most often backwards -- axioms are
| really an end not the starting point.
|
| Here's another view: Euclidean geometry is euclidean because
| the underlying transformation group (the group of those
| transformations which preserve what we want to preserve -- in
| this case the metric) is the euclidean group (the semi-direct
| product of the orthogonal group and translations). This is
| the symmetry that encodes our intuition -- the same intuition
| the kid is using to prove the above result. If we were to
| change the underlying space to the real projective space
| instead of R^2, and instead of choosing to preserve the
| metric we choose incidence and cross-ratio, we'd get a
| different group (GL(3,R)) and different geometry, viz.
| projective geometry.
|
| This is an ancient dialectic that runs within mathematics --
| embodied in modern math by Hilbert on one side (the
| formalist) and Poincare on the other.
| t_mann wrote:
| > Since the triangle got turned completely around, the sides of
| the box must be parallel, so it makes a parallelogram. But it
| can't be a slanted box because both of its diagonals are
| diameters of the circle, so they're equal, which means it must be
| an actual rectangle.
|
| I'd be careful with such "visual proofs", even more so if
| accompanied by such handwavy reasoning. Eg, do we know that both
| diagonals are diameters? Do we know that a parallelogram with
| equal diagonals is a rectangle? While in this case things do work
| out nicely, I'd say this is almost more luck than a real proof -
| it's easy to mistakenly "prove" stuff like Pi=4 with similar
| reasoning. I believe 3B1B even has a video on the topic.
| pringk02 wrote:
| > do we know that both diagonals are diameters
|
| This must be true, because the diagonals are both straight
| lines that go through the centre and are bound by the edges, so
| it follows they must be equal to the diameter of the circle by
| definition.
|
| > Do we know that a parallelogram with equal diagonals is a
| rectangle?
|
| As another commenter points out, this is a theorem you can
| reach for, but proving it by itself is a bit more of a task.
| HarHarVeryFunny wrote:
| > This must be true, because the diagonals are both straight
| lines that go through the centre
|
| How do we know the 2nd diagonal goes through the center ? Is
| it because of the construction by rotation ?
| surajms wrote:
| Yes. So, here when we rotate the triangle, we are
| essentially rotating each of the endpoints. For each
| endpoint, we rotate it by 180 degrees around the line
| segment joining the endpoint and the center. This by
| definition will result in a new position for each endpoint
| that creates a chord (as the two endpoints lie on the
| circle) and passes through the center (we rotated around
| it). A chord that passes through the center is by
| definition a diameter.
| lupire wrote:
| It maybe more clear if you visualize the reflection as a
| pair of perpendicular reflections, first across the
| diameter (which is also across the center) and then
| internally reflecting the diameter (which is again also
| across the center.)
|
| Two reflections with a common fixed point make a rotation
| around that fixed point (angle of reflection is double the
| angle between the reflection axes.). Two perpendicular
| reflections make a 180 degree rotation around the
| intersection of the axes of rotation.
| trueismywork wrote:
| There's nothing handwaving or luck about arguments by symmetry.
|
| Your pi=4 example has more defects than defects in symmetry
| arguments.
| tech_ken wrote:
| I think the point is less to provide a completely proper proof
| of Thales Theorem, and more to demonstrate the fundamental
| principle of what a proof is (an argument to back-up a
| seemingly intractable statement), and how one might construct
| one (use concepts which we already understand, ex. rectangles,
| to create some plausible reasoning). Yes it involves some bad
| habits (relying primarily on visual intuition), but you've got
| to start somewhere. Moreover, the deficiencies of the example
| become the motivators for the next example ("so in the last
| example we did X, but that has problem A, so now we try Y").
| rodneyzeng wrote:
| The normal approach to prove Thale's theorem should be induced
| from the property of central angle being twice of an inscribed
| angle that subtends the same arc. Since a diameter has central
| angle of 180 degrees, its corresponding inscribed angle should be
| half of 180, that is 90 degrees.
| lupire wrote:
| No, because if you do it that way, you wouldn't have Thales's
| Theorem. It would be Thales's Trivial Corollary.
|
| Thales's Theroem is a simpler, easier to prove (as in OP), less
| powerful statement than the inscribed angle theorem.
| rodneyzeng wrote:
| Your second sentence denies the first sentence. The proof of
| the Inscribed angle theorem does not need Thale's Theorem,
| and it is stronger than Thale's Theorem.
| lupire wrote:
| Euclid's proof, with Byrne's beautiful visualization:
|
| https://www.c82.net/euclid/en/book3/#prop31
| ralferoo wrote:
| The corners of the "parallelogram" in the diagram don't touch the
| circle at the top or bottom. So, those two corners wouldn't be
| right angles, but instead would be slightly obtuse - rather like
| this comment!
| tantalor wrote:
| huh
| abecedarius wrote:
| Perhaps this was essentially Thales's own proof:
| https://intellectualmathematics.com/blog/first-proofs-thales...
|
| BTW it's also quite direct using vectors: the legs of the
| triangle are the sum and difference of radius vectors. Take their
| dot product, distribute it, it's zero because radii are the same
| length.
| jacobolus wrote:
| Thales and Pythagoras are quasi-mythical figures, and we don't
| actually know anything concrete about any mathematical
| accomplishments they might have had, which are all apocryphal
| and date from many centuries after their deaths. Greek
| deductive mathematics per se dates from at least a century
| after Thales' time, while many of the basic facts about
| Euclidean geometry were understood in ancient Egypt and
| Mesopotamia long before him, and it is most likely that Thales
| himself never did any of the mathematical or scientific things
| attributed to him.
|
| Viktor Blasjo's speculation that ancient Greeks began with the
| same insight as Lockhart's 7th grade student is plausible but
| is not backed by any evidence whatsoever. (This is an insight
| that many people have had over the centuries, certainly
| including anyone deeply investigating cyclic quadrilaterals,
| but also probably plenty masons or metalworkers working with
| circles and right angles, etc.)
|
| > sum and difference of radius vectors
|
| This is a nice one.
|
| Another way to use Thales' theorem in characterizing a circle,
| without involving the center point, is to start with one point
| _P_ on a circle and a vector _d_ which is a diameter from that
| point to the antipodal point. Then the vector _v_ from _P_ to
| any other point _Q_ on the circle satisfies _v_ 2 = _v_ * _d_ ,
| or equivalently _v_ * ( _v_ - _d_ ) = 0.
| dang wrote:
| Related. Others?
|
| _A Mathematician 's Lament (2002) [pdf]_ -
| https://news.ycombinator.com/item?id=35929333 - May 2023 (47
| comments)
|
| _A Mathematicians Lament [pdf]_ -
| https://news.ycombinator.com/item?id=30829704 - March 2022 (3
| comments)
|
| _A Mathematician 's Lament [pdf]_ -
| https://news.ycombinator.com/item?id=15385104 - Oct 2017 (1
| comment)
|
| _A Mathematician's Lament (2002) [pdf]_ -
| https://news.ycombinator.com/item?id=14331752 - May 2017 (27
| comments)
|
| _A Mathematician's Lament (2002) [pdf]_ -
| https://news.ycombinator.com/item?id=8845507 - Jan 2015 (89
| comments)
|
| _A Mathematician 's Lament [pdf]_ -
| https://news.ycombinator.com/item?id=6994939 - Jan 2014 (1
| comment)
|
| _A Mathematician's Lament (2002) [pdf]_ -
| https://news.ycombinator.com/item?id=6187014 - Aug 2013 (119
| comments)
|
| _New book Measurement by author of Lockhart 's Lament_ -
| https://news.ycombinator.com/item?id=4317199 - July 2012 (4
| comments)
|
| _Mathematician 's Lament: An essay on math education and on how
| we view math_ - https://news.ycombinator.com/item?id=666563 -
| June 2009 (18 comments)
|
| _On Math Teaching: Lockhart 's Lament_ -
| https://news.ycombinator.com/item?id=256176 - July 2008 (21
| comments)
|
| _Lockhart 's Lament: On Mathematics at School_ -
| https://news.ycombinator.com/item?id=130499 - March 2008 (20
| comments)
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