The design of the new math was the fault of mathematicians. I taught my older daughter to read by a combination of look-say and phonics. It went very well. At the end of second grade, she was reading at 7th grade level, and that was a year after I stopped teaching her. However, I also tried to teach her math via the "new math" workbooks. Whether it was Russell's influence or Bourbaki's it based numbers on sets and we had {picture of teddy bear, toy truck, elephant} as examples, i.e. with the curly brackets. The problem with new math is that it is not interesting. It wouldn't be bad if the curly bracket examples lasted only one day, but it had to be dragged out because of the grade level. Even addition and multiplication tables are more interesting than the intersection and union of explicitly given sets. My opinion is that what generates interest in mathematics is geometry in the style of Euclid with theorems whose truth is not obvious and whose proofs are challenging. New math was a disaster with my daughter, because it is boring.

On Tue, 28 Oct 2003, Steve Gray wrote:

Hello John, A few decades ago I visited the A.I. lab at Stanford and Les Earnest showed me around. He demonstrated something he called the "ever-rising tone." It was most impressive. Do you have any idea where I could get a copy of it (in almost any form, audio or digital)?

[Another John asnwering] I don't know. But I once saw it combined in a neat way with one of Escher's infinitely ascending staircases. As one watched a ball bouncing perennially upwards, one heard the ever-rising tone! I'd love to see and hear it again. John Conway

The notes are shown and explained in Godel-Escher-Bach. Just ask Scott Kim to play them for you, the next time Scott and a baby grand piano happen to be nearby. --Ed Pegg Jr, www.mathpuzzle.com

I don't know. But I once saw it combined in a neat way with one of Escher's infinitely ascending staircases. As one watched a ball bouncing perennially upwards, one heard the ever-rising tone! I'd love to see and hear it again.

John Conway

Quoting John Conway <conway@Math.Princeton.EDU>:

I don't know. But I once saw it combined in a neat way with one of Escher's infinitely ascending staircases. As one watched a ball bouncing perennially upwards, one heard the ever-rising tone! I'd love to see and hear it again.

I remember seeing that too, but I'm afraid my reply isn't much help. It may have been one of the web sites devoted te Escherology, and it must have been recent (5 years) or I wouldn't have had audio. - hvm ------------------------------------------------- Obtén tu correo en www.correo.unam.mx UNAMonos Comunicándonos

Steve Gray writes:

A few decades ago I visited the A.I. lab at Stanford and Les Earnest showed me around. He demonstrated something he called the "ever-rising tone." It was most impressive. Do you have any idea where I could get a copy of it (in almost any form, audio or digital)?

These are called Shepard's Tones. Google can find them for you. As the famous Latin saying goes: In Googlis non est, ergo non est. -- Ben Bornstein

Thanks to all who responded. I got way more leads to the "ever-rising tone" than I ever expected. Steve Gray ----- Original Message ----- From: "George W. Hart" <george@georgehart.com> To: "math-fun" <math-fun@mailman.xmission.com> Sent: Monday, October 27, 2003 6:41 PM Subject: [math-fun] Shepard's tones

Steve Gray wrote:

...

A few decades ago I visited the A.I. lab at Stanford and Les Earnest showed me around. He demonstrated something he called the "ever-rising tone." It was most impressive. Do you have any idea where I could get a

copy

...

of it (in almost any form, audio or digital)?

It was probably what is usually called "Shepard's tones" from a 1964 paper by Roger N. Shepard, "Circularity in Judgments of Relative Pitch" Google lists many online versions of it, e.g., http://www.noah.org/science/audio_paradox/

George http://www.georgehart.com/

_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun

I've been to Huntsville, AL, and gone through the used bookstores looking for old physics & math books. They're all in _German_! (i.e., not in French). I can therefore say with certainty that the new math had no effect on the Apollo program (thank god). The major problem with new math (I had some exposure to both old & new math in junior high) is that the teachers didn't know anything about abstract set theory, either, and so it was the blind leading the blind. There is a serious problem with motivation of abstract concepts without concrete examples as rationale. Thus, I had an abstract commutative algebra course in graduate school, but didn't have the corresponding undergraduate background. I followed it 100% as pure abstractions, as none of the examples made any sense to me. I think I got an "A" in the course, but I do _not_ recommend this procedure, as it was "pure" hell. (Teach no abstraction before its time...) At 01:14 AM 10/22/03 -0400, Richard Petti wrote:

O The feds introduced "the new math" into the schools in the 1960s to optimize the education of the more math-able students from whom the nation's technical manpower comes. Fear of Sputnik (read: fear of an untouchable missile base on the moon in the hands of a hostile superpower) caused the feds to take this unorthodox step. The new math taught arithmetic and algebra in terms of sets, mappings, associatitivity, commutativity etc. instead of memorizing computational rules. However, it made life more difficult for abstraction-challenged students, who now had to learn strange abstractions in addition to learning procedural math. The new math was gradually killed in public schools in the 1970s and 1980s. The new math books (the best of which were author by Mary Dolciani) were replaced by procedural math books ('i' before 'e' except after 'c'), the most popular of which were written by John Saxon.

Actually, my latest flame didn't mention math or math education at all. These "science" books are not only devoid of math, they are devoid of *numbers*. They deprive the student of any sense of dimension or scale, even relative. "My" 8th grader and his high school brother found it highly plausible, rather than hilarious, when a cinematic murderous snowman explained his makeup as "genetically altered water". It seems to me that the most glaring ommission from early science education is the notion of dimensioned quantity. How old must you be to understand the difference between a kilowatt and a kilowatt hour? With weights and calibrated balance sticks you could demonstrate even to second graders that Nature multiplies and divides, yet not digitally. Properly built toy cars could demonstrate how momentum and energy scale with mass and velocity, and flexible plastic beams could demonstrate how stiffness and breaking strength scale similarly with breadth, and differently with thickness. And thus how a crack running lengthwise inside a tree limb can lead to it breaking crosswise. Some kids, at least until school destroys it, have "quantitative curiosity". Following are some exercises I proposed in an off-list discussion. How big is a cell, an atom, the moon? Would the moon's orbit fit inside the Sun? Which is faster, the speed of light, or one billion mph? How much would you need to enlarge a grain of salt so its atoms were as large as grains of salt? Suppose one of the atoms was a wayward uranium atom that spontaneously fissioned. Would the mc^2 be enough to raise the grain its own diameter against Earth gravity? The intensity of sunlight hitting Earth is one kilowatt per square meter. What is the total wattage of the sun? If everyone on Earth jumped into the Grand Canyon, would they fit? "Miles per gallon" has the same units as "per acre". What is the physical significance of this (tiny) area? If a block of polyethylene the size of the Moffett blimp hangar were all one molecule, could it stretch to the moon? The Andromeda Galaxy? Everybody should witness this chestnut: Draw a horizontal line on the board about 10 ft long, maybe with vertical ticks on the ends. Tell the class this represents one billion. Ask someone fairly bright to come up and mark off one million. Of course, for the geeks we have: What's the speed of light in furlongs per fortnight? Could that guy who'd walk a mile for a Camel plod a picoparsec for a Pall Mall? Scrooge McDuck's money bin is said to contain three cubic acres. How much is that in square gallons? Lest you find six-dimensional money frivolous, meet nine dimensional pollution: In a Greepeace magazine I saw a photo of a vandalized sewer pipe with a caption crowing that before their sabotage, it had carried "thousands of cubic liters" of pharmaceutical waste to the ocean. We are graduating generations of "science" students who are defenseless against these innumerate morons peddling instant personal validation. --rwg

Some interesting facts: The Greeks used professional human runners as messengers rather than horses; the "Marathon" runner probably perished from wounds or dehydration rather than the 26 mile run, as Greek messengers typically ran more than 26 miles _every day_. As was (re)discovered in the late 1800's, humans can outrun horses, if the distances are long enough. I have a (Dover) translation of Vitruvius, who was a Roman architect in around 100 A.D. time frame, and he mentions Eratosthenes's calculations of the circumference of the Earth, so the spherical nature of the Earth was well-known, at least among the educated Romans & Greeks. The U.S. Navy has apparently discovered a whole host of intact wrecks from the Greek & Roman period on the bottom of the Mediterranean & Black seas. We are on the verge of learning a whole lot more about shipping in ancient times. There's even a scientist who is trying to extract ancient sounds from clay pots that were made on a potter's wheel. His theory is that the sounds modulated the surface of the clay during spinning, so it may be possible to play them back like a record. At 09:39 PM 10/28/03 -0600, mcintosh@servidor.unam.mx wrote:

Some things I would like to know: Was Alexander's "conquest" of the world a leisurely stroll, of did he have to work at it? Is Iraq bigger than California? How long did a sea voyage from Rome to Alexandria take? Is it really true that nobody knows how big a cubit was?

- hvm

[non-math nonsense; delete quick to prevent potential reduction of fun]

=Henry Baker <hbaker1@pipeline.com> Some interesting facts: [...] There's even a scientist who is trying to extract ancient sounds from clay pots that were made on a potter's wheel. His theory is that the sounds modulated the surface of the clay during spinning, so it may be possible to play them back like a record.

Not to detract from the other interesting facts you cite, but I find this very dubious, though it was featured recently in the pop-sci media. Forgive this rant, but I'm finding there's more of this kind of pseudoscience about lately (though more typically in support of sociopolitical agendas). Here, the cited "research" apparently mostly involves manually rubbing a phonograph needle over pots and even painted surfaces and... just listening for... something... that sounds like... something (eg they claimed to just get the word "blue" from some splotch of blue in a large painting)! The potential for delusion in this sort of control-free activity is of course enormous. There are many amusing examples of this (most usually visual), such as that guy who sees clear evidence for tiny fossil unicorns in magnified mineral samples. On closer consideration, you'll find the whole idea is mechanically highly implausible. How did sufficient sound energy get focused in the first place to make the recording clear enough to pick up with a conventional needle (recall those Edison-era phonograph horns?) And since ambient sounds impinge all over the clay, why is the alleged signal spatially localized, instead of "holographically" imprinted? Else what mysterious mechanism somehow causes the potter's massive blunt fleshy finger to act as a precise stylus? And how do these subtle traces persist hanging on the side of a wet glob of mud until, and through, firing? Nah. If it were actually possible to record reliably extractable audio in this way it could be objectively validated by, for example, a simple well-designed double-blind experiment using wholly-mechanized processing to distinguish differently "imprinted" prepared pots. Instead, the advocates leap to seeking permission to scratch up the Mona Lisa in the absurd expectation of somehow "hearing" da Vinci, or his model, or random passersby ("Aren't you done in there yet?" "Klaatu barada nikto!" "Paul is dead."<;-). And why eschew applications? If such effects could be demonstrated at all the potential commercial impact would be significant; look out Napster! Bubble-gum blowing by concert attendees will be forbidden, so as to prevent pirate recordings, etc. Until there's extraordinarily good evidence to support these extraordinary claims, I'm afraid this particular "fact" should be treated (as the Australian Skeptics' site aptly puts it) as "crackpottery".

I said that he was trying; not that he was successful. I do keep hoping that someone will come up with a scheme to extract sound "recordings" of some type from ancient times -- ditto with "photographs".

Here's a way someone *could* have taken a photo, though there's no evidence that anyone ever did: http://www.grand-illusions.com/roman.htm -- Mike Stay staym@clear.net.nz http://www.xaim.com/staym

Hmmm... Very interesting! I hope that someone has checked the Babylonian & Egyptian records -- they were apparently very, very good. Perhaps they did discover Uranus. Although I would imagine that such a discovery would have been considered very important, and therefore been widely disseminated. On the other hand, the Zoroastrians (another group that studied star charts religiously(!) ) were pretty secretive about their findings. By the way, the plaza in front of the new cathedral in downtown Los Angeles _is_ a star chart -- it records the positions of a number of stars and planets on the day that it was consecrated about a year ago. Too bad that the haze & smog of L.A. makes it impossible to compare using the naked eye! At 10:37 PM 10/31/03 -0800, John McCarthy wrote:

Here's another example of something that was never done, although it could have been. Uranus is visible with the naked eye and therefore could have been discovered to be a planet.

In fact it was discovered by Herschel in 1751 who was making a new star catalog and compared his catalog with a previous one prepared by Flamsteed. One star had moved substantially between catalogs and proved to be a planet. Herschel wanted to name it Georgius after the king, but a Greek god was more suitable.

He used a telescope, but it wasn't essential, but the cataloging was.

--- "R. William Gosper" <rwg@tc.spnet.com> wrote:

Actually, my latest flame didn't mention math or math education at all. These "science" books are not only devoid of math, they are devoid of *numbers*. They deprive the student of any sense of dimension or scale, even relative. "My" 8th grader and his high school brother found it highly plausible, rather than hilarious, when a cinematic murderous snowman explained his makeup as "genetically altered water".

Is that related to Penta Water? Penta Water claims to be water imbued with a biologically more compatible molecular structure. It is sold in health food stores for 10 times the price of ordinary bottled H20. Fortunately, the state is not compelling people to buy or drink this product. __________________________________ Do you Yahoo!? Exclusive Video Premiere - Britney Spears http://launch.yahoo.com/promos/britneyspears/

Very cool! Re bolts: I used to work in the clothing industry. "Bolts" of cloth go from nominally 42 inches wide (British wools), to nominally 60 inches wide (American wools). In reality, the usable cloth is around 4-6 inches smaller. Wool is notorious for changing its "size" with temperature and humidity, so clothing manufacturers often put it through a standardization process when it is received which wets it and dries it. The entire bolt is unrolled, goes through the machine, and is re-rolled, during which time the width is continuously recorded & stored in a database for that bolt. I worked on a computer program to virtually "lay out" ("marking") the pattern pieces on the wool to figure out how much cloth a suit would require. Laying out multiple suits at the same time is typically more efficient in terms of fitting the parts together better, but it is more complex to find all the parts after the cutting process and put them together. This process is a 2-D knapsack problem, and is obviously very difficult. Additional constraints involve "one way materials", matching stripes, matching plaids, non-symmetrical stripes, non-symmetrical plaids. It was lots of fun. At 11:59 PM 10/31/03 -0800, R. William Gosper wrote:

My unit-conversion challenges prompted me to write a long-planned Macsyma package. (Macsyma comes with a units package that leaves much to be desired and brings much to be undesired.)

To further show off my package, (c151) uconv(hectare,meter*yard)

12500000 meter yard (d151) ------------------- 1143

a crazy unit of area that was claimed to be used in Japanese cloth trade, due to cutting off metric lengths from bolts woven with the old British loom width. However, Rowlett's definition of "bolt" lists different widths for different fabrics, none of which is a yard.

The problem is actually somewhat more complex; I didn't want to get into all the gory detail. Before cutting, the cloth is folded in two, so that you cut the left & right sleeves simultaneously; the pattern for the collar is cut in half and put against the fold, so that the (one-piece) collar is cut without a seam. "One-way" stripes and asymmetrical plaids don't work this way when folded, and often have to be cut unfolded. The reason you don't cut an unlimited number of suits at the same time, is that the material is also stacked up in multiple plies -- sometimes 100 plies or more. Large electrically powered vertical knives that look like big jigsaws are used to cut these multiple plies, so that you can cut upwards of 50 suits with the same cut. You now have a classic optimization problem: you have a number of suits to cut, the number of each size of suit follows a "normal" distribution, with the largest number being (American) sizes 40 and 42, and smaller numbers of 36 and 48 sizes, etc. If you try to include too many different sizes into the same group, the efficiency of cloth utilization goes up, but the number of plies is reduced, and hence the efficiency of cutting labor goes down. So there is a material $ vs. labor $ tradeoff. At 10:23 AM 11/1/03 -0500, Bernie Cosell wrote:

On 1 Nov 2003 at 7:05, Henry Baker wrote:

...

I worked on a computer program to virtually "lay out" ("marking") the pattern pieces on the wool to figure out how much cloth a suit would require. Laying out multiple suits at the same time is typically more efficient in terms of fitting the parts together better, but it is more complex to find all the parts after the cutting process and put them together.

This process is a 2-D knapsack problem, and is obviously very difficult.

Isn't it more complicated than just a 2-D knapsack problem?

Assuming the metric is "yards per suit", it is hard enough figuring out the optimal answer for, say, "three suits at a time", but if you're trying to actually optimize the assembly-line you have to consider that "four suits at a time" or "five..." or "six..." or "forty..." might result in an overall lower average yards-per-suit, so I'd think that that'd make it harder than just a 2-D knapsack [e.g., to determine that doing 87 suits at a time gives you the *optimal* yards-per-suit]

/Bernie\ -- Bernie Cosell Fantasy Farm Fibers mailto:bernie@fantasyfarm.com Pearisburg, VA --> Too many people, too few sheep <--

Ed Pegg Jr wrote:

My next column for MAA is available for perusal, and last minute corrections/suggestions.

http://www.mathpuzzle.com/Mu.html

Nice column. How should I read the sentence "In the first 100000 tries, the zeta_0 idea is correct a thousand times more than it is wrong"? Do you mean it's correct 50500 times and wrong 49500 times, so #right - #wrong = 1000? Or is #right = 1000*(#wrong)? That's much more impressive. --Michael Kleber kleber@brandeis.edu

Bill, how many photons in a (cd s)? How to deal with the proper units such as : candela (cd) as photons/s [generated by the whole source over 4 pi sr] and lumen (lm) as photons/s [as captured by the receiver's solid angle of observation (sr)] It's like giving different *fundamental* (!?) units to different measures of energy contained in food: <http://www.gov.on.ca/OMAFRA/english/livestock/dairy/facts/92-017.htm#general> Energy A nutrient essential for maintenance, growth, production and reproduction. Energy is required in larger amounts than any other nutrient except water, and is often the limiting factor in livestock production. Gross Energy (GE):. The total combustible energy in a feed, determined by measuring the amount of heat produced when a feed sample is completely burnt in a bomb calorimeter Digestible Energy (DE): Energy that is available to the animal by digestion; measured as the difference between gross energy content of a feed and the energy contained in the animal's feces (gross energy minus fecal energy.) Metabolizable Energy (ME): A measure of the useful energy in a feed. It represents that portion of the feed gross energy not lost in the feces, urine and belched gas. Net Energy (NE): The amount of feed energy actually available for animal maintenance and production. It represents the energy fraction in a feed left after fecal, urinary, gas and heat losses are deducted from the gross energy value of a feed. Total Digestible Nutrients (TDN): A term describing the energy value of feedstuffs, comparable to DE in accuracy. TDN over-estimates the energy value of roughages in comparison to grains. Calorie: A measure of energy; usually expressed as kilocalorie (kcal) or megacalorie (Mcal). 1 cal = the amount of energy required to increase the temperature of 1 g of water 1 degree C. Joule: A unit adopted by Systeme International (SI) for expressing energy. The Joule is more commonly used in Europe than in North America (4.184 J = 1 calorie). W. ----- Original Message ----- From: "R. William Gosper" <rwg@tc.spnet.com> To: <math-fun@mailman.xmission.com> Cc: <rowlett@email.unc.edu> Sent: Saturday, November 01, 2003 8:59 AM Subject: [math-fun] units

My unit-conversion challenges prompted me to write a long-planned Macsyma package. (Macsyma comes with a units package that leaves much to be desired and brings much to be undesired.)

...

"Miles per gallon" has the same units as "per acre".

(c147) uconv(mpg,acre) 12043468800 (d147) ----------- 7 acre

...

What is the physical significance of this (tiny) area?

The cross-sectional area of the thread of gasoline the car would need to continously sip to keep running.

...

What's the speed of light in furlongs per fortnight?

(c148) uconv(c,furlongs/fortnight)

2518256647200000 furlongs (d148) ------------------------- 1397 fortnight or approximately (c149) dfloat(%) 1.80261749978525d+12 furlongs (d149) ----------------------------- fortnight

This package introduces absolutely no floating point approximations.

...

Scrooge McDuck's money bin is said to contain three cubic acres. How much is that in square gallons?

(c150) uconv(3*acres^3,gallons) 2 679899097202688000 gallons (d150) --------------------------- 49

...

Could that guy who'd walk a mile for a Camel plod a picoparsec for a Pall Mall?

What is a parsec?? The Web is rife with units conversion pages that are mostly rubbish. Exception: http://www.unc.edu/~rowlett/units/index.html . This is a "dictionary" of measurements worthy of Weisstein, but without the cupidity of Wolfram and CRC. It defines parsec as the distance at which an astronomical unit (ua) subtends an arc second, and defines ua as the "average" distance to the sun. This contradicts my recollection of parsec defined in terms of the semimajor axis of Earth's orbit. This would also seem a more logical definition of ua, since, as JMC recently reminded us, planetary periods only depend on the major axis. Also, what is meant by "average"? Down in the noise, but relevant to "theoretical" units conversion, is the question of slant height vs altitude: ua cot arcsec or ua/2 csc arcsec/2?

By the way, (c161) uconv(arcsecond,radian) %pi radian (d161) ---------- 648000 To further show off my package, (c151) uconv(hectare,meter*yard)

12500000 meter yard (d151) ------------------- 1143

a crazy unit of area that was claimed to be used in Japanese cloth trade, due to cutting off metric lengths from bolts woven with the old British loom width. However, Rowlett's definition of "bolt" lists different widths for different fabrics, none of which is a yard.

It's very easy to add new units to the package. E.g., after adding the two lines uput('tropicalyear,'canon=365*'days+5*'hr+48*'min+459747/10000*'sec)$ uput('siderealyear,'canon=365*'days+6*'hr+9*'min+954/100*'sec)$ one can ask (c152) uconv(siderealyear,tropicalyear)

35064610600 tropicalyear (d152) ------------------------ 35063251083

The continued fraction of this coefficient is (c153) cf(numfactor(%)) (d153) [1, 25790, 1, 25, 4, 1, 2, 4, 850] which says the difference is very nearly 1 part in 25791.

uput('mhz,'canon='megahertz)$ uput('rpm,'canon='revolution/'min)$ were all that was needed for (c153) uconv(mhz,rpm) (d153) 60000000 rpm

Prefixes and plurals are treated generically:

(c154) uconv(megamicromumble,millimumbles) (d154) 1000 millimumbles

which is not always the right thing!

(c155) uconv(centipede,millipedes) (d155) 10 millipedes

I'll distribute the package when I get this parsec business straightened out, and add a few hundred more units. Who knows, a dumb little application like this might leverage some Macsyma sales. Except Macsyma isn't for sale.-( --rwg

_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun

...

Could that guy who'd walk a mile for a Camel plod a picoparsec for a Pall Mall?

What is a parsec??

Prof. Rowlett kindly answered that it is indeed based on the semimajor axis, which his pages seem to claim is known to +-30m!? Weisstein's parsec page concurs and clearly indicates parsec = ua cot 1" . So (c184) uconv(1.0*picoparsec,mi) (d184) 19.1735 mi (Note how a coefficient of single, double, or multiple precision 1.0 suppresses the big rational coefficient.) So, walkable, but maybe not for a smoker. mkleber>How about uconv(decadent, tridents) ? Rats, I can't

think of any legitimate reason it should recognize tri-.

I have triennium, etc., but not the craziness to genericize those prefixes. On the other hand megacephaly converts to a clinically impressive trillion microcephalies. hgb> I worked on a computer program to virtually "lay out" ("marking") the pattern

pieces on the wool to figure out how much cloth a suit would require. Laying out multiple suits at the same time is typically more efficient in terms of fitting the parts together better,

Mike Sinclair at Microsoft once consulted to an outfit that solved this by analog computation. They laser-cut 1/6 scale patterns from plastic, then shook them for an hour on an inclined table. Finally, a young eavesdropper contributes:

Speaking of units, NASA doesn't seem to understand them. Have a look at http://www.kennedyspacecenter.com/html/saturnVcenter.html:

The amount of power that the Saturn V produced upon blast off (7.5 million pounds of thrust) could light up New York City for 1 hour and 15 minutes!

Wow, they somehow managed to mix up force, power, and energy. I wonder if they used Prentice Hall textbooks to aid them in the calculation.

Your tax dollars at work. --rwg

Actually, this "analog" way doesn't work very well. Aside from the problems of not following other constraints (stripes, plaids, etc.), it doesn't handle the problem of "local optima" very well. For example, if a trouser leg gets into one position vs another trouser leg, then it gets "caught", and the pieces in effect get locked together, no matter how much shakin' is goin' on. This is a standard problem with so-called "simulated annealing" (stimulated annealing?!?), where the problem is handled by occasionally raising the "temperature" to high levels to allow for more global rearrangements. But the convergence rate is very slow on combinatorial problems like this. At 03:21 PM 11/2/03 -0800, R. William Gosper wrote:

hgb> I worked on a computer program to virtually "lay out" ("marking") the pattern

...

pieces on the wool to figure out how much cloth a suit would require. Laying out multiple suits at the same time is typically more efficient in terms of fitting the parts together better,

Mike Sinclair at Microsoft once consulted to an outfit that solved this by analog computation. They laser-cut 1/6 scale patterns from plastic, then shook them for an hour on an inclined table.