[HN Gopher] An Ancient Geometry Problem Falls to New Mathematica...
___________________________________________________________________
An Ancient Geometry Problem Falls to New Mathematical Techniques
Author : theafh
Score : 134 points
Date : 2022-02-08 15:02 UTC (7 hours ago)
(HTM) web link (www.quantamagazine.org)
(TXT) w3m dump (www.quantamagazine.org)
| whatshisface wrote:
| I almost feel bad making such a trivial point in response to such
| a nontrivial article, but they're not solving the ancient problem
| of doing it with a compass and a straightedge (which was proven
| impossible in the late 1800s), they're solving another problem
| that shares the similarity that there's a square and a circle.
| jnsie wrote:
| I agree with you completely. That it was explained in the
| article, does not change the article's title/premise.
| HotHotLava wrote:
| I think this is overly narrow: Anaxagoras considered the
| problem of turning a circle into a square of the same area, and
| these mathematicians shed new light on that problem. He was
| probably thinking of compass and straightedge because it was
| the only language he had for attacking the problem, but it's
| not he published a paper with the precise definition of the
| terms and the theorem he tried to prove.
|
| From what I can tell from a cursory search, there is no
| surviving fragment concerning squares and circles from
| Anaxagoras himself, and the mention on Wikipedia goes back to a
| quote from Plutarch:
|
| > There is no place that can take away the happiness of a man,
| nor yet his virtue or wisdom. Anaxagoras, indeed, wrote on the
| squaring of the circle while in prison.
| thaumasiotes wrote:
| https://en.wikipedia.org/wiki/Squaring_the_circle#History
|
| > It is believed that Oenopides was the first Greek who
| required a plane solution (that is, using only a compass and
| straightedge).
| mlyle wrote:
| That's well explained in the article:
|
| > Because a previous result had demonstrated that it's
| impossible to use a compass and a straightedge to construct a
| length equal to a transcendental number, it's also impossible
| to square a circle that way.
|
| > That might have been the end of the story, but in 1925 Alfred
| Tarski revived the problem by tweaking the rules. He asked
| whether one could accomplish the task by chopping a circle into
| a finite number of pieces that could be moved within a plane
| and reassembled into a square of equal area -- an approach
| known as equidecomposition
| gus_massa wrote:
| The problem is that the title is misleading. It's like an
| article with the title " _New world record in the 100 meters
| dash using a new technique_ " and in the middle of the
| article it explains that it's about bungee jumping a 100m
| fall.
| dfabulich wrote:
| I think it's fair to say that this is an "ancient problem."
| The ancients only had a compass and straightedge, but they
| were asking a general question, "can you square the
| circle?"
|
| They weren't asking "can you square the circle using only
| these two tools currently known to us?"
| gus_massa wrote:
| The old problem is solvable with a compass, a
| straightedge and a rope.
|
| [You wrap the rope around the circle and straight it to
| get a segment of length 2pR, and take the middle point to
| get a segment of length pR. Then use the compass to
| continue it with a segment of length R. And then
| calculate the square root like in
| https://www.geogebra.org/m/edtecfcv to get a segment of
| length sqrt(p)R that is the side of your square. I'm sure
| this was known in ancient times.]
|
| Using only compass and straightedge is more like a
| esthetics decision.
|
| The old problem is difficult (impossible) because you
| have strong restrictions about which points you can draw.
| You have no rope and no magic rule to get any arbitrary
| length.
|
| The new problem is difficult because you must cut one
| figure and rearrange the parts to get the other figure.
|
| They have very different restrictions, in spite both are
| about a circle and a square with the same area.
| mlyle wrote:
| The squaring a circle with compass and straightedge has
| been known to be impossible for 140 years.
|
| We're obviously talking about the version of the problem
| everyone's been working on for roughly the last 100.
| whatshisface wrote:
| Then you must agree that the problem that was solved is
| not in the literal sense of the word "ancient."
| mlyle wrote:
| This is just a further result:
|
| - You can't do it with a compass and a straightedge, like
| the ancients were trying (1880's?)
|
| - You can't do it with scissors, either (mid-20th
| century?)
|
| - But with modern mathematics, and really complicated
| shapes-- you can. (now).
| ruggeri wrote:
| It _might_ be obvious to people who are familiar with the
| problem. In fact, the title confused (and baited) me
| because I was familiar with the problem, but not the new
| formulation.
| paulpauper wrote:
| maybe misleading but still interesting article . Maybe "new
| mathematics offers a solution to an Ancient problem"
| thaumasiotes wrote:
| > He asked whether one could accomplish the task by chopping
| a circle into a finite number of pieces that could be moved
| within a plane and reassembled into a square of equal area --
| an approach known as equidecomposition
|
| Huh. The Banach-Tarski theorem ("you can chop a sphere into a
| finite number of pieces and, by moving them within 3-space,
| reassemble them into a sphere of double the radius") strongly
| suggests this is possible. What's so interesting about the
| revised question?
| mlyle wrote:
| As other people point out, 2-space is not 3-space. Only
| weakened version of B-T work in 2-space (e.g. infinite
| number of pieces).
|
| Tarski, of course, was familiar with his work of the year
| before formulating the B-T paradox when he posed this
| question.
| m00n wrote:
| Actually, a Banach-Tarski-like result is impossible in 2D
| space, since there is a Banach measure (= volume definition
| to all subsets of the plane) that extends the usual volume
| definition (e.g. for circles).
|
| The crucial idea that makes Banach-Tarski work in 3D is the
| insight that the set of rotations around an axis through
| the origin in 3-space has a free subgroup F on 2 generators
| (finite strings of A's, B's and their inverses). From this
| fact the proof is quite easy, but this comment is too small
| for it.
| OscarCunningham wrote:
| Banach-Tarski doesn't work in 2-dimensional space; there
| isn't a finite collection of subsets of the plane which can
| be assembled to make both one disc of radius one and two
| discs of radius one.
|
| I believe that Banach-Tarski would make it much easier to
| disect a sphere and make a cube.
| ummonk wrote:
| Chopping and rearranging something that's the same
| spherical shape (but different size) is different from
| chopping a 2D square and rearranging into a circle.
| Presumably, if it were easy, Tarski himself would have
| shown it, given that he's the one who posed the question.
| [deleted]
| sandebert wrote:
| Relevant Numberphile:
|
| https://youtu.be/CMP9a2J4Bqw
| paulpauper wrote:
| The intellectual level, complexity of research-level math is so
| great these days . Your kid has a greater chance of being a
| multi-millionaire NBA player than being smart enough to
| understand this stuff or do cutting-edge math research, compared
| to something like history or literature. As a field, modern
| mathematics is so far ahead of what laypeople can do but also
| even much of the field itself. It's like, imagine getting a PhD
| in math, which is a hard thing to do, and then multiply by a
| factor 100 in difficulty. Even 18th century math would be a
| challenge for many math grad students. Just crazy
| m00n wrote:
| Sorry, but the breathless way, that maths is often discussed on
| HN, makes me feel uneasy.
|
| It feels strange to see adults that opine on every subject,
| from nuclear fusion energy, to virology and financial markets,
| like they know it all, to suddenly "I was never good at math",
| like a clichee party conversation.
|
| I mean, I get it: It first feels strange and magical, since
| even the explanations of some of the vocabulary take more time
| than we are willing to devote to a single thought. But instead
| of digging in and looking up what "Borel measurable" might
| mean, the HN crowd rather watches the x-th numberphile
| video/emotionalized Quanta blurb.
|
| /rant
|
| More to your points:
|
| > Your kid has a greater chance of being a multi-millionaire
| NBA player than being smart enough to understand this stuff
|
| There are >5000 math phds each year, so no, getting into the
| NBA is harder.
|
| > Even 18th century math would be a challenge for many math
| grad students. Just crazy
|
| Not sure, what this is supposed to mean. Certainly as a math
| grad you should be able to _understand_ 18th century math. Now,
| to _come up_ with the stuff is something else entirely. But I'm
| not sure how many engineers would claim they had discovered the
| telegraph, were they be born instead of Gauss.
| jordan_curve wrote:
| If you looked instead at the number of people who obtain
| tenure at a research university, it would look much more
| comparable to getting into the NBA.
| inglor_cz wrote:
| I was good enough in maths to get a PhD from Commutative
| Algebra, but the really good ones were on another level,
| where you could barely follow their thoughts (especially
| real-time; anything can be attacked with enough patience, but
| it was precisely the _speed_ of their train of thought that
| humiliated you the worst).
|
| People like Erdos were gods in the mathematical universe.
| xyzzyz wrote:
| This is exactly right. I could also get a PhD degree in
| math myself (I dropped out after obtaining Master during
| which I obtained novel results in algebraic geometry), but
| after meeting and interacting with actually smart people,
| it became clear to me that I'm just not nearly on the same
| level. Research level mathematics requires completely
| another level of sheer brainpower that most people don't
| even imagine exists.
| laingc wrote:
| Adding my voice to this too. I have a PhD in Differential
| Geometry and would consider myself to have been a decent
| student and researcher. The "good" people in my field
| were more than a head and shoulders above me, and the
| "great" people were somewhere off in the stratosphere.
|
| The nature of Mathematics is that the potential depth of
| understanding and progress is essentially infinite, which
| frees truly spectacular minds from the constraints they
| would experience in other fields.
| octopoc wrote:
| Is that because the "good" people in your field were just
| way more obsessive about the topic?
|
| I feel like there are some topics that I'm obsessed with
| that I'm so much more informed on than most people in my
| field that I can run circles around them. They would call
| me super smart if the things I'm obsessive about
| mattered. Sometimes they have mattered. But I know better
| than to talk about them at length because people get
| bored.
| sapsucker wrote:
| I 100% agree that some people are innately superior at
| math, the mental arithmetic abilities (at a very young age)
| of human calculators like Von Neumann are proof enough of
| that.
|
| But I also agree with the other poster that it's kind of
| dangerous/distasteful to imply that mathematical ability is
| something that is not necessary to cultivate, or at least
| not worthwhile unless you're the next Galois.
|
| A lot of students are already lacking in grit and give up
| on difficult subjects, not realizing that areas like math
| require a lot of discipline, struggle, and engagement to
| cultivate. This hierarchical nonsense about it only being
| worthwhile for the "chosen few" NBA superstars is not
| productive, especially with Ameria trailing most developed
| nations in mathematical and scientific literacy (which has
| real societal consequences, IMO).
| wolverine876 wrote:
| I saw a study from long ago, maybe the 1980s, which
| researched US and Chinese high school education. As I
| recall, people in the US high schools mostly thought that
| success in education was due to natural talent, while
| people in the high schools in China thought it was
| overwhelmingly due to hard work. The kids in the Chinese
| schools did much better on the tests.
|
| > This hierarchical nonsense about it only being
| worthwhile for the "chosen few" NBA superstars is not
| productive
|
| Agreed. It also takes away the dreams of and
| opportunities from a lot of people.
| thechao wrote:
| I think you're equating "mathematician" with "Fields-medallist-
| adjacent"? As such, I think you'd need to equate such a
| mathematician to the list of "greatest of all time" in
| basketball, who are still alive. I suspect those numbers would
| still tilt in favor of there being more mathematicians than NBA
| players.
|
| On the other hand, I think this is a great way for us science-y
| types to get a good handle on how hard being an NBA player is:
| NBA players are the moral equivalent of near-Fields-medallists;
| _that 's_ how good they are compared to the rest of us.
|
| I'm no mathematical slouch -- I've done grad work in Math,
| taught myself differential geometry, etc.; but it'd be
| fruitless to compare me to Terry Tao. There's really no
| reference for how good he is at math compared to me. I think,
| analogously, you wouldn't be able to compare a college-level
| basketball player to, say, Michael Jordan.
| [deleted]
| mabbo wrote:
| > "I'd bet a beer that you can square the circle, provably, with
| less than 20 pieces," he said. "But I wouldn't bet $1,000."
|
| This is the kind of math that I love because when the results get
| better they get _more_ appealing to the less-math-savvy masses.
| The decomposition of a square into those pieces will quickly
| become a puzzle you give to children and they think it 's hard
| but everyone has seen it before.
| SamBam wrote:
| I think the gif at the top is an approximation of what that
| result would look like -- cutting the square into just 6
| "pieces" to make a circle. But it's absolutely not something
| that can be cut up and given to children.
| riidom wrote:
| It also couldn't be farer away from how I imagined the
| solution would like.
|
| Not that I had anything non-vague in mind, I'm mostly just an
| interested layman, but surely not anything like that gif!
| Truely amazing.
|
| Also I can't even start to get my head behind "Yes there are
| shapes but they are hard to visualize" so how do they even
| work with them?
|
| Or "We have a gap left, of zero area". This is not a gap in
| my book, but you are the experts :) Math at its best.
| mlyle wrote:
| Here, the pieces are fractal holy messes-- not something you
| can make a puzzle out of.
|
| (On the other hand, it is _trivial_ to cheat with small gaps,
| etc, and make such a puzzle).
| kurthr wrote:
| I wonder if a square of the same area is too constraining for the
| axiom of choice?
|
| Perhaps they should try making it into two identical circles or a
| square of twice the area instead?
|
| https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox
| arutar wrote:
| Would you be able to elaborate a bit what you mean here? There
| is no version of Banach-Tarski in two dimensions - you can
| prove that there exists a finitely additive set function which
| is invariant under isometries.
| RcouF1uZ4gsC wrote:
| > The authors show how a circle can be squared by cutting it into
| pieces that can be visualized and possibly drawn. It's a result
| that builds on a rich history.
|
| Ever since I saw the missing square puzzle
| https://en.wikipedia.org/wiki/Missing_square_puzzle
|
| I have been very leery of any geometry proof that requires
| visualization. It is so easy to hide a small difference.
| mannykannot wrote:
| To be clear, the proof is not dependent on visualization.
| mlyle wrote:
| (But is more easily visualized than previous proofs).
| mc32 wrote:
| [totally naive -no maths background] What happens when you make a
| circle so large that the boundary in small enough arcs is
| practically straight?
| toppy wrote:
| I think EVERY circle fulfills this condition ;)
| thaumasiotes wrote:
| A circle of radius 0 may not.
| bell-cot wrote:
| For this sort of math (theoretical), "practically straight" is
| never straight enough.
|
| And your approach is close enough to calculus that "totally
| naive..." seems a tad too modest.
| yongjik wrote:
| Math isn't concerned with "practically straight", so it won't
| make a difference.
|
| For example, consider the number 2 and 2.000000000001. To
| engineers and scientists, these two numbers are basically the
| same, and there's no practical difference. To a mathematician,
| the former is an integer, and the latter isn't - you could add
| as many zeros to the second number as you want, and the
| difference won't go away, unless you add an _infinite_ number
| of zeros, at which point it becomes identical to 2.
| gus_massa wrote:
| The construction uses the radius of the circle as a parameter.
| If the radius is bigger, then each piece is bigger.
|
| Imagine that you have this construction drawn as a svg file in
| the computer, a really big screen, and you can use the zoom.
| The size of the pieces will match the size of the circle.
| syki wrote:
| The radius of the circle doesn't come into play. This is
| because instead of enlarging the circle, as you say, what one
| does is narrow in on smaller and smaller sections of the
| circle. Given any circle one can narrow in on a small enough
| piece that, essentially, when looking at it, it will be almost
| straight.
|
| This is true for almost all curves you can think of and draw
| and is the basis of calculus. Calculus is the study of
| functions whose graph locally looks like a straight line.
|
| Inscribe a 30 sided polygon inside a circle of radius 10 cm.
| Visually you'll find it hard to see the difference between the
| polygon and the circle. Using the formula for the area of a
| triangle you can calculate the area of the inscribed polygon
| very easily. This provides an approximation to the area of the
| circle.
|
| Now do this for a 40 sided polygon. Then a 50 sided polygon. A
| pattern will emerge and one then sees that the limit, which is
| what happens as the number of sides gets larger and larger
| without bound, is the familiar formula for the area of a
| circle. This is how you can prove what the formula for the area
| of a circle is. You can think of a circle as an infinite sided
| regular polygon.
| posterboy wrote:
| I think the point of the comment was precisely that I cannot
| simply let n be infinite. In fact, there's an old joke where
| that's the punchline.
| whatshisface wrote:
| I think you could get close by making a lot of pie slices,
| lining them up in a row, and then flipping every other one
| upside-down. Then you'd be able to put them back together into
| a rectangle.
|
| The problem with our plan, I guess, is that you'd always be
| close at every pie slice size, never exactly there. These guys
| have figured out how to do it exactly, with a finite number of
| slices, instead of approaching it in the limit of infinity
| slices.
| lscharen wrote:
| I get a kick out of these kinds of pure math problems where
| there's such a large gap between what is provable and what might
| be the best answer.
|
| "We can prove it can be done with 10^200 pieces, but it can
| probably be done with less than 20".
|
| Close enough.
| cyberbanjo wrote:
| In terms of all the choices, it's a pretty small sliver
| gilleain wrote:
| A great example of such a range of bounds is the problem that
| Graham's number was an upper bound for :
|
| https://en.wikipedia.org/wiki/Graham%27s_number
|
| "Thus, the best known bounds for N* are 13 <= N* <= N''." Where
| N'' is ... really hard to type, it's so large!
| tromp wrote:
| It's not hard to type in lambda calculus [1]:
|
| (l (l 2 1 (l 1 (l l 1 2 (l 1)) (l l 5 (2 1)) 3) 1) (l l 2 (3
| 2 1))) (l l 2 (2 (2 1)))
|
| [1] https://mindsarentmagic.org/2012/11/22/lambda-graham
| dmonitor wrote:
| it is when your keyboard doesn't have a lambda key
| richardfey wrote:
| It's just an ALT + 955 away!
| phkahler wrote:
| Wait a minute. I thought there was a simple non-existence proof
| something along the lines of: Given the square, you'll need to
| cut pieces with the outer arc(s) of the circle. When you cut any
| length of convex arc, you also create a piece with and equal
| length of concave arc which will then require that much more
| convex arc to fill in the end. In other words, the circle
| requires a certain amount of arc length, and every time to create
| a piece to fit that you add a requirement for an equal amount of
| arc somewhere else.
|
| I suppose this could be resolved by some kind of fractal, but
| that's going to have an infinitely long perimeter.
| Sharlin wrote:
| Yes, the pieces here are highly nontrivial in shape, as
| discussed in the article. But more well-behaved than in earlier
| results.
| jessriedel wrote:
| Yes, as discussed in the article, the shapes are not piecewise
| smooth curves (i.e., not the sort of shapes you can construct
| by making a finite number of straight and smoothed cuts).
| Furthermore, as also mentioned, the areas of the shapes are
| non-measurable, so the perimeter is probably non-measurable
| too.
| [deleted]
| hw-guy wrote:
| This is similar to the claim that an orange, say, can be cut into
| pieces that can then be put together to make two oranges. It
| turns out some of these pieces would be infinitesimal, and hence
| smaller than the atoms making up the orange (or whatever). While
| such a result may be satisfying to a theoretical mathematician,
| the engineer in me recoils.
| Someone wrote:
| Mathematically, it's quite different. The Banach-Tarski paradox
| (https://en.wikipedia.org/wiki/Banach-Tarski_paradox) changes
| the _volume_ of the objects. That's requires some of the prices
| to be immeasurable.
|
| It also is about a 3D sphere, and the strong form (cutting a
| sphere in _finitely_ many parts and reassembling those into two
| equal-sized spheres) doesn't work in 2D or 1D (in contrast, in
| 3D, _five_ pieces suffice. I don't know whether that is a tight
| bound)
| thaumasiotes wrote:
| > The Banach-Tarski paradox
| (https://en.wikipedia.org/wiki/Banach-Tarski_paradox) changes
| the _volume_ of the objects. That's requires some of the
| prices to be immeasurable.
|
| It changes the volume by a discretionary amount; you can
| create two spheres of the same size as the original sphere,
| or 500 spheres of the same size as the original sphere, or
| you can create one sphere of double the radius [= four times
| the size] of the original sphere.
|
| I see no reason to believe that you couldn't also make one
| cube of equal volume to the original sphere?
| OscarCunningham wrote:
| It is a tight bound.
|
| http://matwbn.icm.edu.pl/ksiazki/fm/fm34/fm34125.pdf
| https://www.irregularwebcomic.net/2339.html
| nmilo wrote:
| But here the pieces aren't infinitesimal. They're fractals, but
| still measurable.
| mrob wrote:
| Fractals are still cheating IMO, because fractals have
| infinitely small features. Whenever you use infinity you can
| get all kinds of crazy results. It's like the geometric proof
| that pi=4:
|
| Draw a circle of diameter 1
|
| Draw a square touching it on all sides, perimeter 4
|
| Cutting at right angles to the existing edges, cut smaller
| squares out of all the corners so they touch the circle
|
| Perimeter remains 4
|
| Repeat this corner cutting infinity times
|
| Perimeter of the cut square (4) matches the circumference of
| the circle (pi)
|
| pi = 4
|
| Unlike traditional geometry, it's just abstract symbol
| manipulation with no relevance to real shapes.
| mb7733 wrote:
| That proof is just plain incorrect, though. It will break
| down when trying to prove this statement:
|
| >Perimeter of the cut square (4) matches the circumference
| of the circle (pi)
|
| Calculus will show that the area of the fractal approaches
| the area of the circle. But it will not show that the
| perimeter of the fractal approaches the circumference of
| the circle. It remains 4 at every step in the iteration, so
| the limit is still 4.
| contravariant wrote:
| The pieces in this particular example seem to be quite a bit
| better behaved. In fact they're measurable.
| xoxxala wrote:
| My High School Geometry teacher caught a bunch of us goofing off
| and talking in class, so he assigned us the squaring the circle
| problem (with compass and straightedge) as an extra assignment.
| Said if we solved it, he would give us an A for the semester. We
| had no idea and worked really hard on that for a few weeks before
| he told us.
| arutar wrote:
| I'm not sure why this is not mentioned in the article, but there
| is nothing special about circles and squares (or 2 dimensions,
| for that matter). If anything, phrasing it like this gives the
| (misleading) impression that somehow features of squares and
| circles are important!
|
| The authors proved [1, Thm. 1.3] that given any two sets in R^d
| with equal non-zero measure and boundaries that are "not too
| horrible" (i.e. box / Minkowski of their boundaries less than d),
| one can cut one of the sets into finitely many Borel pieces and
| rearrange them (i.e. apply isometries in R^d) to obtain the other
| set.
|
| You can also guarantee that the pieces have positive measure
| under a mild technical assumption.
|
| [1] https://arxiv.org/pdf/2202.01412.pdf
| EGreg wrote:
| So we can't claim it's impossible to square a circle now?
| Glyptodon wrote:
| Is the mentioned proof about not being able to create a
| transcendental number length segment with a compass and straight
| edge, I think I'm missing the boundaries of how constructions are
| permitted within the proof. Is this effectively because you need
| more than 2 linkages to translate curved motion to linear motion?
| (As I'd assume any device that converts circular motion to linear
| motion would produce linear motion in ratios of pi.)
| Jtsummers wrote:
| https://en.wikipedia.org/wiki/Constructible_number
|
| That is the definition of "constructible". In order to perform
| (with straightedge and compass) the squaring of the circle, you
| need to construct a line of length sqrt(pi) in a finite number
| of steps. However, since sqrt(pi) is a transcendental number,
| that's impossible.
| kmote00 wrote:
| Frustrating article. Suggests that somebody has come up with a
| way to cut jigsaw pieces differently to arrange a rectangular
| puzzle into a perfect circle. The solution is so simple that the
| shape of the pieces can actually be described and visualized. And
| then, after all this tantalizing buildup, the big reveal is...
| hidden behind a paywall. :(
| gilleain wrote:
| What's interesting to me is how many of the ancient problems
| involve using compass and straightedge. Recently I have been
| trying to draw Islamic geometric patterns (or other tilings, like
| quasitilings) using compass and ruler, and it can be really
| difficult!
|
| I can kind of see, though, why considerations of what integer
| ratios are 'good' for such diagrams and questions like angle
| bisection or intersections between circles and lines become
| interesting topics. It can really affect how easy or hard it is
| to draw such a diagram
| whatshisface wrote:
| Compasses and straightedges were their attempt to distill the
| nature of plane geometry down to its essential and simplest
| form (lines and circles).
| ogogmad wrote:
| It can even be reduced to Clifford algebra very cleanly:
| https://en.wikipedia.org/wiki/Conformal_geometric_algebra
|
| There is something very "right" about it.
| riidom wrote:
| A note to the circles: It is not so much about circles, but
| rather "Some points which share the same distance to another
| point X"
|
| If you follow a manual of how to construct something with
| compass and straightedge, the job of the circles is often
| only to intersect with something else, and these points of
| intersection are of actual interest (as far as I remember).
| gilleain wrote:
| Exactly this. There are many patterns that can be
| constructed by drawing a regular array of circles, then
| connecting various intersection points with lines, then
| erasing the original circles.
|
| As it happens, I sometimes find that the same drawing can
| be achieved by a simpler construction path that involves
| (say) only midpoints of squares, which makes life a lot
| easier.
| thaumasiotes wrote:
| How do you find the midpoint of a square without drawing
| circles?
| whatshisface wrote:
| Draw the diagonals.
| abecedarius wrote:
| I wonder, though, is it not a coincidence that they're
| practical tools to do rather precise multi-step
| constructions? (E.g. Durer wrote a whole book about type
| design by those methods.) And with all the geometric algebra
| in Euclid, did they ever use them for calculations that
| aren't originally geometrical? Would we know?
| whatshisface wrote:
| I believe that ancient geometry was used for governance and
| engineering. However the compass and straightedge had an
| element of abstractness or deliberate simplified
| impracticality even back then: they had rulers and strings,
| and could have practically used them as well.
| gilleain wrote:
| Agreed, I am sure that the drive for axiomisation of geometry
| drove a lot of this interest.
|
| All I really mean is that actually using these tools for an
| artistic, constructive purpose gives me a feel for why these
| problems might of been of interest. Of course, without
| knowing much about the history of mathematics this far back,
| I cannot be sure.
___________________________________________________________________
(page generated 2022-02-08 23:01 UTC)