November 2018 - mathfun@mailman.xmission.com

[math-fun] Ford Circles addendum
by Bill Gosper 27 Nov '18

27 Nov '18

There's a simple connection with that notorious function which is continuous only at the irrationals. gosper.org/rectarith12.pdf, pp 10,11. —rwg

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[math-fun] Ford circles & Euclid's Algorithm
by Henry Baker 26 Nov '18

26 Nov '18

I think that the pretty pictures of Ford circles can be "improved" into an actual Euclidean algorithm. First, draw the Ford circles in the *complex* plane (NOT the Ford spheres discussed in Ford's article). Each of these Ford circles can be mapped into another via *inversion* (complex 1/z) through a third circle. Represent the circles using Schwerdtfeger's circle representation: http://en.wikipedia.org/wiki/Generalised_circle More details about how this might be done from my July 8, 2012 post to math-fun: ---- I've been hacking a new version of the Common Lisp & Maxima "rationalize" function which produces the worst rational approximation to a complex number that falls within epsilon of that complex number. In the case of Maxima, the epsilon is set by the global variable "ratepsilon". The traditional Euclidean algorithm for this process focuses on the number itself (which I call c), and only later worries about the epsilon and the radius of convergence. My algorithm starts with the _circle_ whose center is the given complex number c and whose real radius is r; we can denote this circle by (c,r). Since translation and inversion in the complex plane both take circles to circles (more accurately, "generalized circles" to "generalized circles"), we can perform a Euclidean procedure on the _circle_ (c,r) itself. At each stage we keep track of the transformations (translations & inversions), so that we can recover the original circle, if necessary. We first look at the given circle (c,r). If there is a (Gaussian) integer point within this circle, then we are done, because such a point is rational, so this is our answer. If we compute cr=round(c) (= round(realpart(c))+i*round(imagpart(c))), then this number cr is (one of) the closest integer points to c. If cr is "inside" the circle (c,r), then we are done. Now if r>sqrt(2)/2, then cr _has_ to be inside the circle (c,r). So we now consider the case where cr is not inside (c,r). After translating the origin to cr, we know that this new center is within sqrt(2)/2 from the origin -- i.e., inside the unit circle. We then invert the given circle in the unit circle, which will put the center of that new circle outside the unit circle. Furthermore, the radius of that new circle will grow bigger. We can now compute the closest integral point to the center of that new circle, and so forth. At each of these Euclidean steps, the circle radius grows larger, and must eventually enclose an integral point, at which time we invert the sequence of transformations to compute the final rational answer. This answer will be rational, because the integral point was transformed by inversions and translations by (Gaussian) integers. This discussion is not a _proof_, but shows how the algorithm was conceived. Here's the algorithm in Maxima: euclid(zc,m):= block([q,iqmat], /* zc is input circle in Schwerdtfeger's representation */ /* m accumulates transforms; m starts out as the 2x2 identity matrix */ /* Original circle is always transpose(m^^-1).zc.conjugate(m^^-1) */ q:cround(center(zc)), if is(inside(q,zc)) then m.matrix([q],[1]) else (iqmat:matrix([1,q],[0,1]), euclid(transpose(iqmat.imat).zc.conjugate(iqmat.imat),m.iqmat.imat))); This produces is a column vector matrix([num],[denom]); the result is num/denom. euclid(gcircle(%pi,1.0e-7),ident(2)) produces [ - 122453 ] [ ] [ - 38978 ] = 122453/38978, which is within 10^-7 of pi. (%i13) abs((-122453)/(-38978)-%pi),numer; (%o13) 3.9724384226502707E-8 We use Hans Schwerdtfeger's representation for generalized circles as 2x2 Hermitian matrices. See this Wikipedia article for more details. http://en.wikipedia.org/wiki/Generalised_circle /* We test for "inside" by substituting the point into the equation. */ inside(w,c):=is(equation(c,w)<=0); Although very elegant, there is a problem with Schwerdtfeger's representation for circles: the squared radius of the circle is computed by the difference of two large quantities, so when we approach the limit of machine precision, we may inadvertently end up with a negative number, and hence an _imaginary_ radius! Perhaps this problem can be ameliorated with a slightly different representation.

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[math-fun] a real, non singular-value, "doable" EllipticK
by Bill Gosper 26 Nov '18

26 Nov '18

(Big deal. The singular ones are probably 0% of the doables.) EllipticK[( 6561 (Sqrt[2] + 5^(1/4))^8 (-1 + Sqrt[5])^4 (1 + Sqrt[5])^16)/( 1048576 (1 + Sqrt[2])^8 (Sqrt[2] + Sqrt[5])^8 (1 + Sqrt[ 10])^4)] == ( 4 (1 + Sqrt[2])^2 (1 + 5^(1/4))^4 (Sqrt[2] + Sqrt[5])^2 (1 + Sqrt[ 10]) Sqrt[2/\[Pi]] Gamma[1/4]^2)/( 9 (Sqrt[2] + 5^(1/4))^2 (-1 + Sqrt[5])^2 (1 + Sqrt[5])^6) EllipticK[ 1 - (6561 (Sqrt[2] + 5^(1/4))^8 (-1 + Sqrt[5])^4 (1 + Sqrt[5])^16)/( 1048576 (1 + Sqrt[2])^8 (Sqrt[2] + Sqrt[5])^8 (1 + Sqrt[ 10])^4)] == ( 8 (1 + Sqrt[2])^2 (1 + 5^(1/4))^4 (Sqrt[2] + Sqrt[5])^2 (1 + Sqrt[ 10]) Sqrt[2/\[Pi]] Gamma[1/4]^2)/( 45 (Sqrt[2] + 5^(1/4))^2 (-1 + Sqrt[5])^2 (1 + Sqrt[5])^6) EllipticK[( 6561 (Sqrt[2] + 5^(1/4))^8 (-1 + Sqrt[5])^4 (1 + Sqrt[5])^16)/( 1048576 (1 + Sqrt[2])^8 (Sqrt[2] + Sqrt[5])^8 (1 + Sqrt[10])^4)]/ EllipticK[ 1 - (6561 (Sqrt[2] + 5^(1/4))^8 (-1 + Sqrt[5])^4 (1 + Sqrt[5])^16)/( 1048576 (1 + Sqrt[2])^8 (Sqrt[2] + Sqrt[5])^8 (1 + Sqrt[ 10])^4)] == 5/2 The complementary parameter likewise factors into binomials: 1 - (6561 (Sqrt[2] + 5^(1/4))^8 (-1 + Sqrt[5])^4 (1 + Sqrt[5])^16)/( 1048576 (1 + Sqrt[2])^8 (Sqrt[2] + Sqrt[5])^8 (1 + Sqrt[10])^4) == ( 324 Sqrt[2] (Sqrt[2] + 5^(1/4))^4 (1 + Sqrt[5])^6)/((1 + Sqrt[ 2])^4 (1 + 5^(1/4))^12 (Sqrt[2] + Sqrt[5])^4 (1 + Sqrt[10])^2) I think Ramanujan actually understood why. —rwg

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[math-fun] eccentricity eccentricity
by Bill Gosper 26 Nov '18

26 Nov '18

Do you share my annoyance that the formula for the a-by-b ellipse, √(1-(b/a)²), only works for a ≥ b? Like turning an ellipse 90º turns it into a hyperbola? Is there a formula for √(1-(Min[a,b]/Max[a,b])²) nicer than √(1 - E^(-2 Abs@Log[b/a]))? Some arctrig(trig)? True|False puzzle: An ellipse is a special case of an ellipsoid. My answer: .si espille derevoc-elbuod a tub ,oN The three eccentricities of the axis-aligned cross-sections of an a by b by c ellipsoid are not independent, and the asymmetry of the eccentricity formula makes things weird. If a ≥ b ≥ c, the eccentricities are ba := √(1-(b/a)²) cb := √(1-(c/b)²) ca := √(1-(c/a)²) with ba^2 = (ca^2 - cb^2)/(1 - cb^2) i.e. cb^2 = (ca^2 - ba^2)/(1 - ba^2) but then ca^2 = ba^2 + cb^2 - ba^2 cb^2 For each ellipse shape, we can imagine a different π: π(eccentricity) = circumference/major axis, π(0) = π, π(1) = 2, π(𝜀) = 2 EllipticE(𝜀²). This would make a nice alternative quantification of ellipticity if EllipticE weren't such a pain. It is possible to choose semiaxes a > b > c (the interesting spinning body!) so that at least two of the elliptic integrals come out in closed form. E.g., eccentricities ca = (√2 - 1)², cb = 4^⅝ (√2 - 1), and hence ba = √(6√2 - 3)/4^⅝ determine π ratios π( (√2 - 1)²) = 2 EllipticE( (√2 - 1)⁴) = (2-√2) π^(3/2)/(8 ¼!²) + 4√(2/π) ¼!² π(4^⅝ (√2 - 1)) = 2 EllipticE(8 - 4 √2) = (2-√2)√π³/(4 ¼!²) + 4 ¼!²/√π π(√(6√2 - 3)/4^⅝) = 2 EllipticE[⅜(4 - √2)] = ? I haven't looked very hard for a nonspheroidal ellipsoid with all three (π)s in closed form. —rwg What kind of prigs would deprive us of vulgar fractions 3/2 and 5/4? 5/4 is traditional in carpentry. And 3/2 dates back millennia as "sesqui". (I know of a school that still marks kids off for "improper fractions".) Not that anyone has said otherwise, but there are infinitely many "non singular-value" k for which EllipticK(k) (and EllipticE(k)) come out in closed form. E.g., EllipticK[1/2 (1 - Sqrt[2])]/EllipticK[1/2 (1 + Sqrt[2])] == 1/3 (I + Sqrt[2]) (instead of the square root of an integer) EllipticK[1/2 (1 - Sqrt[2])] == (8 2^(1/4) (1/8)! (3/8)!)/(3 Sqrt[\[Pi]]) EllipticK[1/2 (1 + Sqrt[2])] == (8 2^(1/4) (1/8)! (3/8)!)/((I + Sqrt[2]) Sqrt[\[Pi]]) (These would be more interesting if they were real. Unless they can't be. Which would be interesting.)

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Re: [math-fun] Messages in pi
by Dan Asimov 25 Nov '18

25 Nov '18

Indeed it is possible. (Even if pi is not normal to any base.) Now I'm curious how *probable* it is, or complementarily, how probable it is that at least one finite string is missing from an infinite string of digits chosen at random. When doing calculations like this I tend to get stuck when trying to account for substrings that overlap. At least for finite calculations. Hmm, the chance of missing a given single digit d is 1 - (1/10)^oo. The chance of missing a single pair of consecutive digits (d, e) is ??? Maybe it's best to *estimate* the chance of N consecutive digits chosen independently at random having every string of length k at least once (for N = N(k) >> 0), and then let N —> oo ??? —Dan David Wilson wrote ----- Presuming pi is normal in all bases, aren't all possible (finite) messages somewhere in there? -----

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Re: [math-fun] Mathematics of SETI protocols ?
by Keith F. Lynch 25 Nov '18

25 Nov '18

Henry Baker <hbaker1(a)pipeline.com> wrote: > The problem is: how do you establish communication with someone > you don't know, and someone who doesn't speak the same language ? > Somehow, when we are babies, we manage to figure out how to > communicate with our parents. Good question, though I'm not sure that it's math. I've never heard of parents proving their intelligence to their newborn, or vice versa, by generating a prime number sequence or the digits of pi. Probably it's mostly pointing at things while speaking their name. Some linguists suggest that much of language is genetic, as proven by the fact that all human languages allegedly have certain features in common, rather than being random points in the space of all possible languages. Decades ago, I asked Chomsky, in person, whether Lojban, a constructed spoken language, follows his rules for all natural languages. He didn't know enough about Lojban to answer. (Some people are fluent in Lojban. As far as I know nobody has tried to teach it to a very young child. It would probably be unethical to raise a child exclusively in that language.) People also have the ability to recognize other languages, despite utterances in them not usually being prefaced by "I'm about to speak in ..." in a language they know. As for SETI, the best suggestion I've heard is to send a semi-prime (product of two primes) number of on-off pulses which when assembled into a rectangle in one of the only two possible ways form a coherent image. Sending the sequence of prime numbers would communicate only the fact of intelligence, not any content. Sending the digits of pi would at best do the same. But in which base should pi be sent? And maybe the aliens consider 1/pi, 2pi, or sqrt(pi) the constant, and maybe they use continued fractions, Egyptian fractions, or something we never thought of instead of a base number system. > Each TM can't assume that there is only 1 other TM that wants to > establish communication -- i.e., TM#1 may get confused if TM#2 and > TM#3 both try to establish communication with TM#1 at the "same" > time. One rather silly suggestion is that there are so many aliens trying to communicate with us that they are jamming each other's signals. > Does any of this discussion ring a bell? Surely, someone must have > studied this sort of problem before? Plenty of good science fiction (and of course far more bad science fiction) has dealt with this. H. Beam Piper's 1957 short story "Omnilingual" is about human archaeologists on Mars, and how they used a Martian periodic table of the elements and other Martian scientific literature as a Rosetta stone to gradually figure out the extinct Martians' written language. Robert Forward's 1980 novel _Dragon's Egg_ portrays the interaction between the inhabitants of a neutron star and a human mission in orbit around it. It's complicated by the fact that the metabolism and thinking speeds of the neutron-star creatures are millions of times faster than ours. During the brief human mission, they evolve from savagery to a civilization much more advanced than ours. Ted Chiang's 1998 novella "Story of Your Life" depicts an alien language so alien that the consciousness of any person learning it comes unbound to time. They remember their future along with their past. The title comes from the chief scientists remembering their marriage to each other and the whole of the life of their daughter, despite neither having happened yet. Even if you think that's absurd, their discussion of variational principles in physics as an alternative to cause and effect is worth reading. It was made into a movie, _Arrival_, in 2016. Carl Sagan's novel 1985 _Contact_ is a classic, though not very original, SETI novel. Unoriginal except for a subplot about the aliens having discovered messages hidden in the base-eleven digits of pi, which was unfortunately left out of the 1997 film of the same name. I find it an interesting question whether evidence for such a message in pi can ever overcome the possibilities that it's either an astonishing coincidence or someone hacking the computer that's doing the calculation. Presumably if there really was such a message, it would have to be from God, rather than from advanced aliens. In the 1974 novel _The Mote in God's Eye_ by Larry Niven and Jerry Pournelle, people visiting an alien planet find an automated museum intended to provide knowledge to reboot civilization. They correctly conclude from that that the alien's civilization is prone to repeatedly crashing long and hard, something the aliens didn't want known. Math can be something aliens communicate with us about. And of course mathematical physics and information theory are essential for designing any method of communicating over interstellar distances. It's recently been suggested that we should look in the optical rather than the microwave part of the spectrum. With higher frequencies, there's linearly more quantum noise (i.e. fewer photons per joule of energy), but quadratically better ability to aim a beam from a given aperture. Stars are much noisier in the optical than in the microwave range, but are easily nulled out with a good interferometer. Anyhow, stars have a very broad spectrum, but an optical signal can rise above the noise by having either a very narrow bandwidth or a very short pulse time, and if the former, by being in the middle of one of the star's dark Fraunhofer lines. Also, the optical part of the spectrum has far more bandwidth, hence channel capacity. With just a few kilowatts total power, everyone on Earth could send an email the size of this one every day to someone near any nearby star. Greg Egan takes this to an extreme, by having people communicate over interstellar distances by gamma ray laser. They sometimes even transmit the locations of every atom in their body, as a form of travel.

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[math-fun] Dandelin spheres video
by Dan Asimov 25 Nov '18

25 Nov '18

Just happened on this YouTube video: ----- What does genius look like in math? Where does it come from? (Dandelin spheres) ----- by Grant Sanderson, in his very good series of math videos on YouTube, called "3Blue1Brown". It's particularly notable for great illustrative graphics (this one and his others). (After going through the proof that the locus of constant sum of distances to two foci in the plane is the same curve as a conic section, he gets into a discussion of genius in mathematics which, though interesting, is not the reason I'm mentioning this video.) —Dan

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Re: [math-fun] Mathematics of SETI protocols ?
by Keith F. Lynch 25 Nov '18

25 Nov '18

D J Makin <makinmagic(a)tiscali.co.uk> wrote: > Not really ?Mathematics? as such - but how do we do it when the part > or parts of the electro-magnetic and/or chemical and/or even base > quantum or other physical manifestations of existence the aliens use > to communicate or just plain sense, is unknown ? We have to make assumptions. For instance to assume that the aliens think somewhat similarly to how why do. This seems plausible if they evolved on a planet similar to ours. We have the experience of lots of very different cultures here on our planet, albeit all of the same species. And of knowing how differently individuals in our own culture think from each other, including beliefs (I know a guy, otherwise apparently sane, who believes everything Alex Jones says), ethics, ways of learning, decisions about what to learn, and time preference. For instance I find choosing to smoke cigarettes incomprehensible. And I continue to be baffled by why anyone would append a complete copy of the message they're responding to to their reply to it (doing that may be useful in SETI, but it's annoying here). I am not a visual thinker. Many of my dreams are non-visual. I was able to learn SQL in less than an hour, but was unable to learn OpenGL at all, except by rote memorization of a few things that worked. I am routinely beaten at rapid visual pattern-matching games such as Jungle Speed by small children. We also have an enormous variety of animals with very different senses and behaviors we can study. We can think about what their society might be like if they were to evolve intelligence. I'm very close to believing that "intelligence" is an incoherent concept, and that there are only skills at various tasks, not an overriding Spearman's G. There certainly appears to be little correlation, except at the very low end, between skill at taking IQ tests and success in life. We don't yet have the technology to detect interstellar signals of plausible strength in any form but electromagnetic. For instance the weakest gravitational wave events that LIGO can detect are powerful enough that if they were visible light, we could read by them. Our best neutrino detectors are better, but not by much. And of course there may be forms of radiation (or non-radiation means of long distance communication) which we haven't discovered yet. It's an interesting question whether our progress in math, physics, and technology took the only possible path. For instance could aliens discover branches of math we never thought of, but completely fail to discover counting, addition, and multiplication? Could there be an alien OEIS with no overlap with Sloane's OEIS? In Harry Turtledove's 1985 short story "The Road Not Taken," aliens attempt to invade our planet with muskets and other technology that's about what we had around 1500 -- except that they do have a star drive, based on principles that we never stumbled into, since those principles are so different from the ideas of Newton, Watt, etc., and their successors. And in Poul Anderson's 1960 novel _The High Crusade_, aliens with advanced technology attempt to invade Earth in the 13th century, but English soldiers end up stealing their starship and taking over the whole galaxy, since the English were much better than them at leadership and the non-technological aspects of warfare, and since the galactic civilization had a major organizational problem which feudalism was a good solution for. Probably neither of those stories is plausible. But if there are creatures living in very un-Earthlike environments, they are likely to learn physics in a different order, and to come up with very different technologies. For instance in _Incandescence_, a 2008 Greg Egan novel, non-technological aliens living in a hollow asteroid in a close orbit around a black hole figure out general relativity with some very simple experiments. The good news is that advanced technology means we can simultaneously scan a large part of the EM spectrum for a large number of ways in which a signal might plausibly differ from noise. Not in every possible way, of course, but certainly the two most obvious, a very narrow band with modulation or very short regularly-spaced pulses with gaps, and maybe a hundred others. (Well, the most obvious to *me*, maybe not to them.) Of course the aliens would have to be intending to be understood. For one thing, unless they're using utterly enormous amounts of power, their signal must be aimed at our solar system rather than being omnidirectional. For another, the most efficient way to use spectrum, i.e. maximum channel capacity for a given bandwidth and power, results in a signal indistinguishable from noise to anyone who doesn't know the details of the modulation. For instance compare the size of the specifications for NTSC (the analog television standard formerly used in the US) and ATSC (the digital television standard that replaced it). I think there's little doubt that aliens could have figured out NTSC given nothing but a very strong and persistent signal in that format. Even now, aliens on a planet around Aldebaran may be watching I Love Lucy, though there's an even chance they have the image flipped left-to-right. But I doubt anyone, even here on Earth, could make any sense of an ATSC signal unless they read the manual (or, obviously, used a television built by someone who read the manual). Actually, there's one obvious alternative to a signal aimed directly at us. The aliens could put one or more giant opaque rotating rings around, and perpendicular to, their star, with patterns of immense holes in them. Their star would then appear, from vast numbers of other solar systems (though not nearly to all of them) to be blinking in specific patterns. Such rings could be extremely thin, hence not need much matter. To nearby solar systems, it could even be read by eye. It's an interesting question what the ancient Greeks would have made of Sirius blinking in a prime-number pattern. There is a star which is blinking in an unexplained way, though not with any obviously intelligent pattern. See https://en.wikipedia.org/wiki/KIC_8462852 (Perhaps we're in an unfortunate direction in which the star is blocked by two such rings, which is why we can't make any sense of the pattern. If so, further observations should allow us to deconvolve the two signals.) Another possibility, which I posted about here on April 1, 2015, is that instead of transmitting non-stop for perhaps billions of years (we've had radio for only one of the 45 million centuries that our solar system has existed) the aliens would leave a message. I suggested they might plant a tuning-fork-containing titanium or tungsten sphere in bedrock on our planet, sound from which could be detected after an earthquake by a phased array of microphones embedded in the top of the bedrock. Has anyone tried listening for such a thing? Or working out whether such a sphere could survive for long? On the other hand, if the aliens could come here long ago, would they have bothered to leave a message and go away, or would they have colonized the place? The Fermi paradox says that if aliens exist in our galaxy, and neither progress nor capital accumulation ends, that even if they're only a tiny fraction of the age of the galaxy older than us, they should have reached everywhere already, and their presence should be as blindingly obvious as the presence of Pilgrims at the (alleged) first Thanksgiving was to the Indians there. One obvious answer, albeit one that tends to be unpopular with my fellow science fiction fans, is that there are no intelligent aliens, never were, and never will be. It's possible that some step in the formation of intelligent life is so extremely unlikely that it only happened once in our entire universe. If it's true that our universe is one of a vast number of universes with different laws of physics, and that most sets of laws of physics are incompatible with the evolution of intelligent beings, and that there are many different ways in which such a universe can fail to be compatible with the evolution of intelligent beings, one would expect that most habitable universes would be near the edge of habitability. (Most points inside an N-dimensional sphere are very close to the surface if N is large.) Perhaps "habitability" is the wrong word. I don't mean to imply that it's very difficult for life to exist in our hypothetical lonely universe. Indeed, it may someday spread from Earth to fill billions of galaxies. I mean that the laws of nature have to be such that intelligent life has to not merely be capable of persisting once it exists, but has to be capable of evolving or in some other way coming into existence in the first place. There are of course countless other possible explanations for the Fermi paradox. Perhaps technological progress always levels out not much beyond where we are now. It may already be happening here. The world of 2018 is much more like the world of 1968 than the world of 1968 was like the world of 1918. Except in computers and communication, there's really been rather little progress in the past fifty years, at least compared to the previous fifty, or the fifty before that. At the other extreme, we may have already long since reached the omega point, and may be living in a simulation, perhaps as an experiment to learn what our distant ancestors were like.

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Re: [math-fun] Ford circles (Henry Baker)
by Henry Baker 25 Nov '18

25 Nov '18

Re quaternions: Hamilton himself studied CF's of quaternions, but I don't think he got very far (by today's standards). I'm curious about CF's of square matrices -- particularly CF's of matrices like unitary matrices (whatever CF's for those might mean!) --- More Fords (circles, that is) in your future: https://arxiv.org/pdf/1508.01373.pdf http://oro.open.ac.uk/35905/8/__userdata_documents4_ctb44_Desktop_amer.math… https://arxiv.org/pdf/1412.1457.pdf https://sites.math.washington.edu/~morrow/336_14/papers/bo.pdf http://lup.lub.lu.se/luur/download?func=downloadFile&recordOId=8892215&file… At 04:38 PM 11/24/2018, Fred Lunnon wrote: > The later part generalises his method to complex numbers; >so what about rational approximation to quaternions? WFL > >On 11/24/18, John Golden <goldenj(a)gvsu.edu> wrote: >> I made a quick GeoGebra applet to help me visualize these terrific Ford >> circles: https://www.geogebra.org/m/v4qn96yr >> >>> Date: Sat, 24 Nov 2018 08:09:25 -0800 >>> From: Henry Baker <hbaker1(a)pipeline.com> >>> Subject: [math-fun] Ford circles >>> >>> I just discovered this paper on "Ford circles" (1938) for fractions and >>> continued fractions. >>> >>> Very very cool & elementary. Suitable for high school students. >>> >>> Either >>> >>> https://trungtuan.files.wordpress.com/2017/06/ford1938.pdf >>> >>> or >>> >>> https://www.maths.ed.ac.uk/~v1ranick/papers/ford.pdf

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Re: [math-fun] Ford circles (Henry Baker)
by John Golden 25 Nov '18

25 Nov '18

I made a quick GeoGebra applet to help me visualize these terrific Ford circles: https://www.geogebra.org/m/v4qn96yr > Date: Sat, 24 Nov 2018 08:09:25 -0800 > From: Henry Baker <hbaker1(a)pipeline.com> > Subject: [math-fun] Ford circles > > I just discovered this paper on "Ford circles" (1938) for fractions and continued fractions. > > Very very cool & elementary. Suitable for high school students. > > Either > > https://trungtuan.files.wordpress.com/2017/06/ford1938.pdf > > or > > https://www.maths.ed.ac.uk/~v1ranick/papers/ford.pdf

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