July 2014 - mathfun@mailman.xmission.com

[math-fun] "Two lines intersecting at angle t
by Bill Gosper 25 Jul '14

25 Jul '14

in the unit circle confine two opposite arcs whose average is t" is equivalent to Cos[1/2 (-a + b - c + d)] == (Cos[a - b] - Cos[b - c] - Cos[a - d] + Cos[c - d])/ (2 Sqrt[(1 - Cos[a - c]) (1 - Cos[b - d])]) where a,b,c,d are angles on the circle. Is there a 3D analog of this theorem? Something like a cone with apex within the unit sphere cuts out two patches whose areas average to some function of the cone's solid angle? A couple of pictures look unpromising. --rwg Mma weirdness: In[1264]:= FindInstance[-1 == Sign[Sin[a - b] + Sin[b - c] + Sin[c - a]], {a, b, c}, Reals] Out[1264]= {{a -> 8/5, b -> 2, c -> 223/10}} OK, but In[1265]:= FindInstance[-1 == Sign[Sin[a - b] + Sin[b - c] + Sin[c - d] + Sin[d - a]], {a, b, c, d}, Reals] takes forever(?) finding nothing(?), even though In[726]:= Table[Sign[Sin[a - b] + Sin[b - c] + Sin[c - d] + Sin[d - a]] /. {a -> RandomReal[], b -> RandomReal[], c -> RandomReal[], d -> RandomReal[]}, {foo, 9}] Out[726]= {1, -1, 1, -1, -1, 1, 1, 1, -1} Presumably 50%.

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Re: [math-fun] SETI
by Henry Baker 22 Jul '14

22 Jul '14

If what you are saying is correct, SETI couldn't recognize the signals that NASA sends to its own deep space probes. I don't know for sure whether SETI is looking for deep space probe-type signals, but it would be an interesting question for the SETI folks. At 06:07 AM 7/22/2014, Warren D Smith wrote: >SETI is the Search for Extraterrestrial Intelligence. > >Much of it might be better re-titled, "search for stupidity" because >the most obvious ways aliens might try to detect us, rely on the >stupidity of humanity. >If you want to transmit information optimally your >signal will bear a great resemblance to "thermal noise" >because that is the signal with the greatest entropy for a given power level. >However, being as humanity is stupid, it creates a lot of signals that are >not at all like that. > >In particular, high-def television (USA) is incredibly stupidly done, >plus involves >a high amount of periodicity at the screen and line refresh rates. >Similar remarks could be made about, e.g. radar systems. >TV and radio commercials that keep being re-broadcast >could be spotted as a statistically amazing exact repeat of the same stuff. >Of course, an alien would be detecting that stuff polluted by a huge >amount of noise, >so it would not be an exact repeat. But using FFTs one could detect >statistically amazing periodic components of signals, and also do >autocorrelations to detect statistically amazing high correlations between >F(t) and F(t+delay). This all is computationally cheap (NlogN time). >If however you were to examine some time-window and compare it with a >different time-window to detect a repeat that way, you could get a lot >more sensitivity to that repeat >than from any FFT-based autocorrelation, but the compute cost would be >much greater >to try many possibilities. > >So, far as I can tell, what I just said summarizes a large set of SETI efforts. >A paper on this topic: > http://setiathome.berkeley.edu/sah_papers/status_of_ucb_seti_efforts_2011.p… > >If anybody were intentionally trying to communicate with us, a good >way seems to be >using short powerful optical laser pulses. Optical is far superior to radio >because it can be directed more precisely. Laser pulses from the largest lasers >available today could easily outshine our sun during the short >(nanosecond scale) pulse, >as viewed from far away in the shining direction, and if we only >observe in a small >band near the laser frequency, then the outshining of the sun can be made an >enormous factor. > >A paper on that: > http://seti.harvard.edu/oseti/oseti_apj_preprint.pdf >which was published as > Howard, Andrew W.; Horowitz, Paul; Wilkinson, David T.; Coldwell, >Charles M.; Groth, Edward J.; Jarosik, Norm; Latham, David W.; >Stefanik, Robert P.; Willman, Alexander J., Jr.; Wolff, Jonathan; >Zajac, Joseph M.: >Search for Nanosecond Optical Pulses from Nearby Solar-Type Stars, >Astrophysical Journal 613,2 (Sept 2004) 1270-1284. > >The Berkeley paper includes "figure 1" showing 112 locations of about >330 unexplained statistical outliers their "astropulse" program >detected, on the sky. (Seeks 1 microsec to 1 millisec long radio >pulses and tries to discard ones from terrestrial sources.) The >surprising(?) thing: >these locations are claimed to be concentrated near the Galactic Plane, in a way >that is statistically significant. If this concentration cannot be >explained as terrestrially caused, then I guess that "proves" that >whatever triggers their "I found something" detector, >is coming mainly from our galaxy. And if, furthermore, natural >sources could not have produced such pulses, they would have to be >artificial. > >The Harvard-Princeton optical pulse paper found only one mysterious >optical pulse event >after using several telescopes each >with about 1 meter diameter totaling about 2400 hours observing time >observing 13000 nearby stars. Just one pulse was >simultaneously detected by more than 1 telescope. This >event came from the K class late dwarf called "HIP 107395" and >occurred on 17 sept 2003. Unfortunately >their hyperaccurate atomic clock wasn't working at this time, hence >they were unable to confirm their two telescopes got the pulse within >e.g. the same >microsecond; they only had millisecond accuracy. Hence this detection is not a >statistically sure thing, albeit it still seems suspicious -- why the >hell should such a star be emitting a microsecond long (or less) >optical pulse (or two such, separated by a millisecond or less)? > >-- >Warren D. Smith >http://RangeVoting.org <-- add your endorsement (by clicking >"endorse" as 1st step)

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[math-fun] SETI
by Warren D Smith 22 Jul '14

22 Jul '14

SETI is the Search for Extraterrestrial Intelligence. Much of it might be better re-titled, "search for stupidity" because the most obvious ways aliens might try to detect us, rely on the stupidity of humanity. If you want to transmit information optimally your signal will bear a great resemblance to "thermal noise" because that is the signal with the greatest entropy for a given power level. However, being as humanity is stupid, it creates a lot of signals that are not at all like that. In particular, high-def television (USA) is incredibly stupidly done, plus involves a high amount of periodicity at the screen and line refresh rates. Similar remarks could be made about, e.g. radar systems. TV and radio commercials that keep being re-broadcast could be spotted as a statistically amazing exact repeat of the same stuff. Of course, an alien would be detecting that stuff polluted by a huge amount of noise, so it would not be an exact repeat. But using FFTs one could detect statistically amazing periodic components of signals, and also do autocorrelations to detect statistically amazing high correlations between F(t) and F(t+delay). This all is computationally cheap (NlogN time). If however you were to examine some time-window and compare it with a different time-window to detect a repeat that way, you could get a lot more sensitivity to that repeat than from any FFT-based autocorrelation, but the compute cost would be much greater to try many possibilities. So, far as I can tell, what I just said summarizes a large set of SETI efforts. A paper on this topic: http://setiathome.berkeley.edu/sah_papers/status_of_ucb_seti_efforts_2011.p… If anybody were intentionally trying to communicate with us, a good way seems to be using short powerful optical laser pulses. Optical is far superior to radio because it can be directed more precisely. Laser pulses from the largest lasers available today could easily outshine our sun during the short (nanosecond scale) pulse, as viewed from far away in the shining direction, and if we only observe in a small band near the laser frequency, then the outshining of the sun can be made an enormous factor. A paper on that: http://seti.harvard.edu/oseti/oseti_apj_preprint.pdf which was published as Howard, Andrew W.; Horowitz, Paul; Wilkinson, David T.; Coldwell, Charles M.; Groth, Edward J.; Jarosik, Norm; Latham, David W.; Stefanik, Robert P.; Willman, Alexander J., Jr.; Wolff, Jonathan; Zajac, Joseph M.: Search for Nanosecond Optical Pulses from Nearby Solar-Type Stars, Astrophysical Journal 613,2 (Sept 2004) 1270-1284. The Berkeley paper includes "figure 1" showing 112 locations of about 330 unexplained statistical outliers their "astropulse" program detected, on the sky. (Seeks 1 microsec to 1 millisec long radio pulses and tries to discard ones from terrestrial sources.) The surprising(?) thing: these locations are claimed to be concentrated near the Galactic Plane, in a way that is statistically significant. If this concentration cannot be explained as terrestrially caused, then I guess that "proves" that whatever triggers their "I found something" detector, is coming mainly from our galaxy. And if, furthermore, natural sources could not have produced such pulses, they would have to be artificial. The Harvard-Princeton optical pulse paper found only one mysterious optical pulse event after using several telescopes each with about 1 meter diameter totaling about 2400 hours observing time observing 13000 nearby stars. Just one pulse was simultaneously detected by more than 1 telescope. This event came from the K class late dwarf called "HIP 107395" and occurred on 17 sept 2003. Unfortunately their hyperaccurate atomic clock wasn't working at this time, hence they were unable to confirm their two telescopes got the pulse within e.g. the same microsecond; they only had millisecond accuracy. Hence this detection is not a statistically sure thing, albeit it still seems suspicious -- why the hell should such a star be emitting a microsecond long (or less) optical pulse (or two such, separated by a millisecond or less)? -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] Weierstraß ℘ and Pisano solutions
by Bill Gosper 22 Jul '14

22 Jul '14

rwg> Ironically, http://functions.wolfram.com/EllipticFunctions/WeierstrassP/16/ has the multiple and half "angle" formulæ, so FunctionExpand should evaluate the explicit general formula exactly, possibly providing nicer recurrences than I have given here.<rwg For starters, the double "angle" formula, for general congruent # n, gives this amazingly simple x2 = Out[1115]= (n^2 + x1^4)/(2 x1 Sqrt[x1^4 - n^2]) In[1116]:= % /. n -> 5 /. x1 -> 5/2 Out[1116]= 41/12 There was a time when some of us wondered if this was the only solution to x^2±5 = ⃞. In[1117]:= %1115 /. n -> 5 /. x1 -> % Out[1117]= 3344161/1494696 Here go n = 6 and 7: In[1118]:= %1115 /. n -> 6 /. x1 -> 5/2 Out[1118]= 1201/140 In[1119]:= %1115 /. n -> 6 /. x1 -> % Out[1119]= 2094350404801/483435791720 In[1120]:= Sqrt[%^2 + {6, -6}] Out[1120]= {2405943600001/483435791720, 1727438169601/483435791720} In[1121]:= %1115 /. n -> 7 /. x1 -> 5 Out[1121]= 337/120 rwg> Maybe even elementary closed forms.<rwg Well, simple as it is, I don't expect to see anyone writing the nth iterate of %1115. --rwg

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[math-fun] The third and second congruent numbers (i.e. 7 and 6)
by Bill Gosper 22 Jul '14

22 Jul '14

We seek all rational x for which both x² ± n are square for some given integer n. E.g., for n = 5, x could be 41/12, because √((41/12)² ± 5) = 10/3 ± 3/4. These x are also the semihypotenuses of all the rational right triangles of area n. E.g., the triangle with sides 3/2, 20/3, 41/6 = √((3/2)²+(20/3)²) has area 5. Those n, (5, 6, 7, 13, 14, ...) for which this is possible are called "congruent". The case n=5: xⱼ = r₂ⱼ, the entries of (positive) even subscript in the sequence r₀, r₁, r₂, ... = ∞, 5/2, 41/12, 11285/1562, 3344161/1494696, 44572169525/7118599318, ... where rⱼ₊₁ =5 |(3 rⱼ - sⱼ 2 √(rⱼ⁴ - 25))/(4 rⱼ² - 25)| and s₁, s₂, ... = +,-,+,+,-,+,-,-,+,... . The wrong choice of sign will step backwards instead of forward: rⱼ₋₁ =5 |(3 rⱼ + sⱼ 2 √(rⱼ⁴ - 25))/(4 rⱼ² - 25)|. The correct choice of sⱼ is whichever produces the larger denominator, which can be predicted: sⱼ = sgn((j α mod 1) - 1/2) where the constant α := ℘⁻¹((5/2)²/4,5²/4,0)/(2 ω(5²/4,0)) ~ -0.372715613160377781284923995013737 , which is(!) all the digits you'll ever need. (ω(g₂,g₃) := the first half-period of ℘(z,g₂,g₃).) According to N Bickford, sⱼ is also the sign of yⱼ in j-tuplication (xⱼ², yⱼ) := j ⊗ (x₀², y₀) on the elliptic curve y² = (x² - 5) x² (x² + 5). An explicit general formula (per N Bickford) for rⱼ in terms of r₁: rⱼ = 2 √℘(j ℘⁻¹(r₁²/4,n²/4,0),n²/4,0) where yⱼ² = (rⱼ² - n) rⱼ² (rⱼ² + n) and n is a congruent number. E.g., for n=5, r₁ = 5/2, and j=1, Mathematica gives for r₂ N[2 Sqrt[WeierstrassP[ 2 InverseWeierstrassP[{(5/2)²/4, {5²/4, 0}], {5²/4, 0}]]] 3.416666666666666 + 0.*I, i.e. 41/12. Bickford also adds that yⱼ = 4 ℘'(j ℘⁻¹(r₁²/4,n²/4,0),n²/4,0) The case n=7: xⱼ = r₂ⱼ, the entries of (positive) even subscript in r₀, r₁, r₂, ... = ∞, 5, 337/120, 708965/246431, 23058557761/4231560720, 19986637352562725/341046511360991, 216673159777897263181777/46857162265896537676440 where rⱼ₊₁ = |(24 rⱼ - sⱼ 5 √(rⱼ⁴ - 49))/(rⱼ² - 25)| and s₁, s₂, ... = +,+,-,-,+,+,-,-,+,+,-,-,+,+,-,-,+,... . As usual the wrong choice of sign will step backwards instead of forward. sⱼ = sgn((j β mod 1) - 1/2) where the constant β := ℘⁻¹(5²/4,7²/4,0)/(2 ω(7²/4,0)) ~ -0.203443585375537507164711731996275 . For n=7, r₁ = 5, and j=1, Mathematica gives for r₂ N[2*Sqrt[WeierstrassP[2*InverseWeierstrassP[25/4, {49/4, 0}], {49/4, 0}]]] Out[1073]= 2.8083333333333336 + 0.*I I.e., 337/120. The case n=6 appears to differ: x₁, x₂, ... = 5/2, 1201/140, 7776485/2639802, 2094350404801/483435791720, 18428872963986767525/4433082615462019402, 5405257799550679424342410801/1793801602727678650180032420,... all give squares for xⱼ² ± 6, without the intervening odd-index values to simplify the recurrence xⱼ₊₁ = √|-3600 + 49 xⱼ² + 100 xⱼ⁴ - sⱼ 140 xⱼ √(xⱼ⁴- 36)|/(4 xⱼ² - 25) where s₁, s₂, ... = +,-,+,-,+,-,+,-,+,+,-,+,-,+,-,+,-,+,+,... sⱼ = sgn((j γ mod 1) - 1/2) where the constant γ := ℘⁻¹((5/2)²/4,6²/4,0)/(2 ω(6²/4,0)) ~ -0.445514890183131231186597446651214 The explicit general formula gives, for x = 5/2, n=6, j=2: N[2*√WeierstrassP[2*InverseWeierstrassP[25/4/4, {36/4, 0}], {36/4, 0}], 22] 8.578571428571428571429 In[1085]:= Rationalize[%] Out[1085]= 1201/140 It's a wonder that Rationalize works, given the behavior of ContinuedFraction: In[1086]:= ContinuedFraction[ 2*\[Sqrt]WeierstrassP[ 2*InverseWeierstrassP[25/4/4, {36/4, 0}], {36/4, 0}]] During evaluation of In[1086]:= ContinuedFraction::noterms: 2 Sqrt[WeierstrassP[2 InverseWeierstrassP[{25/16,35/32},{9,0}],{9,0}]] does not have a terminating or periodic continued fraction expansion; specify an explicit number of terms to generate. >> Out[1086]= ContinuedFraction[ 2 Sqrt[WeierstrassP[ 2 InverseWeierstrassP[{25/16, 35/32}, {9, 0}], {9, 0}]]] In[1087]:= ContinuedFraction[ 2*\[Sqrt]WeierstrassP[ 2*InverseWeierstrassP[25/4/4, {36/4, 0}], {36/4, 0}], 69] During evaluation of In[1087]:= ContinuedFraction::terms: Warning: ContinuedFraction only obtained 6 of 69 requested terms using precision 1104. Try increasing $MaxExtraPrecision. >> Out[1087]= {8, 1, 1, 2, 1, 2} In[1090]:= Block[{$MaxExtraPrecision = 99999}, ContinuedFraction[ 2*\[Sqrt]WeierstrassP[ 2*InverseWeierstrassP[25/4/4, {36/4, 0}], {36/4, 0}], 69]] During evaluation of In[1090]:= ContinuedFraction::terms: Warning: ContinuedFraction only obtained 6 of 69 requested terms using precision 100104. Try increasing $MaxExtraPrecision. >> Out[1090]= {8, 1, 1, 2, 1, 2} In[1089]:= ContinuedFraction[1201/140] Out[1089]= {8, 1, 1, 2, 1, 2, 7} Interestingly, putting j=1/2, 3/2, ..., in the general formula gives 3 √2, 99 √2/47, 134691 √2/23183, 4972102467 √2/2802064609, 8785723105593027 √2/64067278329889, ... for the "missing" alternating values. Perhaps these can denest the radical in the recurrence. ContinuedFraction, of course, also lies that they are non-periodic. Ironically, http://functions.wolfram.com/EllipticFunctions/WeierstrassP/16/ has the multiple and half "angle" formulæ, so FunctionExpand should evaluate the explicit general formula exactly, possibly providing nicer recurrences than I have given here. Maybe even elementary closed forms. --rwg

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[math-fun] Pairs (f,g) of maps [n] --> [n] such that f g = g f
by Joerg Arndt 21 Jul '14

21 Jul '14

Hope it is OK to occasionally(!) raise an OEIS related question. This is about A181162 ( https://oeis.org/draft/A181162 ). I have no better idea than the author: exhaustive search. It seems almost certain that this (nice) problem has been investigated somewhere. Any pointers? Best, jj

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[math-fun] Robots With Origami Wheels
by Henry Baker 21 Jul '14

21 Jul '14

FYI -- It would be interesting to try a wheel like this on a unicycle... http://spectrum.ieee.org/automaton/robotics/robotics-hardware/robots-get-fl… Robots Get Flexible and Torqued Up With Origami Wheels By Evan Ackerman Posted 16 Jun 2014 | 18:05 GMT Origami, the art of folding pieces of paper to create shapes, is an appealing concept for robotics because you can transform two dimensional materials into three dimensional structures that are inherently flexible, or, as a roboticist would say, "deformable." What's more, structures that fold and unfold enable all kinds of interesting functionality that would otherwise only be possible with systems that are much more complex. The approach can be particularly useful in designing wheels for robots, and earlier this month at the IEEE International Conference on Robotics and Automation (ICRA) two research groups presented origami-inspired wheel systems that allow mobile robots to be nimbler and stronger. One of the groups was from Seoul National University's BioRobotics Laboratory, led by Professor Kyu-Jin Cho. Researchers there have designed a clever robotic wheel based on one of the most famous origami patterns, the magic ball pattern. The wheel they created can change its radius by deforming its shape. This is a useful trick to be able to perform, since a wheel with a large radius is better at climbing over things, while a wheel with a smaller radius is better at squeezing under things, as the robot demonstrates: The wheel and hooks together can deform from a minimum diameter of 55 millimeters to a maximum diameter of 120 millimeters, which is a substantial range, especially considering that the transformation only requires one single actuator per wheel. "Fabrication of Origami Wheel Using Pattern Embedded Fabric and Its Application to a Deformable Mobile Robot," by Dae-Young Lee, Ji-Suk Kim, Jae-Jun Park, Sa-Reum Kim, and Kyu-Jin Cho from Seoul National University, was presented this month at ICRA 2014 in Hong Kong. In another project presented at ICRA, researchers at the Harvard Microrobotics Lab, led by Professor Robert Wood, teamed up with the group from Seoul National University to design origami wheels that can automatically expand and shrink. The goal in this case wasn't just allowing the robot to climb over some things and squeeze under other things. Since there's a direct relationship between the radius of a wheel and the torque that it can exert, a wheel that can deform can also act as an automatic, continuously variable transmission. First, a word (make that words) about transmissions: in the most general sense, a transmission is the thing that turns the power output of a motor into some combination of speed and torque. Usually, the way a transmission works is that you can trade torque for speed (or vice versa) while optimizing the efficiency of your motor: in automobiles, a low gear gives you lots of torque but limits your top speed, while a high gear gives up torque to let you go faster. Most automobiles have discrete gearboxes, where you can choose from some number of fixed gear ratios. Inevitably, this means that most of the time, the transmission is not optimized for what you want to do. A continuously variable transmission, on the other hand, offers an infinite number of gear ratios over a fixed range. Unlike most automobiles, most robots don't bother with transmissions at all, because they're heavy and complicated. This means that robots have to be designed to either move fast (which doesn't require a lot of torque), or haul stuff (which does), but they're not great at doing both. The researchers designed an origami wheel that can do both, by simply changing its radius, and thanks to its origami design, it does this by itself. It's a completely automatic continuously variable transmission, which allows a small robot to either move fast or haul stuff without requiring any additional complexity beyond an origami wheel: There are three robots in the video below. There's one with big fixed wheels (high speed, low torque), one with small fixed wheels (low speed, high torque), and then one with origami wheels that can expand and shrink. No adjustment is necessary: if the origami wheels have too much load on them to rotate, they stall, and as the wheel hubs continue to turn, the wheels collapse, which increases their torque until they can move: To reiterate, this transmission is both continuously variable and completely automatic. The wheel can adopt any effective gear ratio in the range between its minimum and maximum diameters, and it does this passively like a spring, as it responds to loads on it by shrinking its diameter until it achieves the maximum diameter at which it can consistently rotate. The cost of the origami wheel is 70 percent of the cost of a fixed diameter wheel, less whatever it costs to hire someone who knows what they're doing to fold it up for you. The researchers suggest that it would also be ideal for applications where weight and volume are issues, like interplanetary rovers, as the wheel can be folded up and then deploy itself. "A Passive, Origami-Inspired Continuously Variable Transmission," by Samuel M. Felton, Dae-Young Lee, Kyu-Jin Cho, and Robert J. Wood, from Harvard University and Seoul National University, was presented this month at ICRA 2014 in Hong Kong.

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[math-fun] The lissajoke rolling on Bob Hearn's floor plus Euler disk breakthrough
by Bill Gosper 21 Jul '14

21 Jul '14

Thane>I took a video of a metal version of Bill Gosper's lissajoke rolling on the floor at Bob Hearn's house ( http://gosper.org/lissajoke.gif ). Completely coincidentally, Bob Hearn is narrating its ambiguous rolling behavior as I'm pushing it around on the floor. Anyway, here is the video https://www.youtube.com/watch?v=7YpYCgRt9Qc&list=UUbBGpH5hVMFF23WjuaFzjcw The ambiguous rolling effect is quite noticeable when I push it farther away. ------------- Alan Adler's rejoinder suggests rpm outweighs viewing distance. And mentions something much more newsworthy: AJA>I posted the Lissajous video here http://youtu.be/Rli5x91Jg64 with this description: This is a rotating Lissajous shape designed by mathematician Bill Gosper. The shape is four inches in diameter and is rotating about its axis. Guess the orientation of the axis (either horizontal or vertical). There's a good chance that you'll guess wrong. There is another Gosper Lissajous shape here https://www.youtube.com/watch?v=7YpYCgRt9Qc I posted my 5:54 Euler disc run here http://youtu.be/N_bMbuEQ6wM <AJA --rwg

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[math-fun] sesqui-imponderables
by Bill Gosper 20 Jul '14

20 Jul '14

If a chicken and a half lays an egg and a third ... Hey, wait a minute, why isn't "chicken and a half" plural? And which came first, the chicken and a half or the egg and a half? --rwg

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Re: [math-fun] sesqui-imponderables
by Bill Gosper 20 Jul '14

20 Jul '14

On 2014-07-20 05:41, Allan Wechsler wrote: > Committing my usual error of wading into a frivolous conversation with my > Serious Hat on... > > Number in grammar is not a mathematical concept; it's a purely formal one. > The "number" of a noun phrase, and hence the verb form you have to pick to > agree with it, is determined by the formal structure of the phrase and of > its component words, *not* by the meaning of the phrase. Now, different > people can have form-agreement rules that differ in various details. For > me, "A chicken and a half lays ..." feels wrong (even without the > subjunctivizing "if"), so the phrase *is* plural for me. If the subject > isn't plural for Gosper and Andy, it must be because they are (in some > sense) parsing it differently, so that it has singular *structure*. My > guess is (and I can't be certain, because I don't have their linguistic > intuitions to guide me) that for them, the and-a-half construction converts > the noun phrase from a "count noun" to a "mass noun"; mass nouns always > parse as singular in American English. Not intuition--convention. Although it is too DWIMmy too tell for sure, Google seems to find many more "... lays"s than "... lay"s. --rwg > > > On Sun, Jul 20, 2014 at 5:24 AM, Bill Gosper <billgosper(a)gmail.com> wrote: > >> On Sun, Jul 20, 2014 at 1:18 AM, Andy Latto <andy.latto(a)pobox.com> wrote: >> >> On Sat, Jul 19, 2014 at 12:49 PM, Adam P. Goucher <apgoucher(a)gmx.com> >> wrote: >> >> 1. ((a (chicken)) and (a (half))), not (a ((chicken) and (a (half)))). >> >> >> That doesn't explain why it isn't plural; compound subjects of the >> form "A and B" are normally plural. >> >> Frog and Toad are Friends. >> >> Andy >> >> So >1 isn't essential for plural, else >> "Two half-wits writes a grammar book, ..." >> >> --rwg

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