September 2005 - mathfun@mailman.xmission.com

[math-fun] Re: Packing circles into a finite rectangle
by James Propp 10 Feb '06

10 Feb '06

>> If you have circles of radii 1/1, 1/2, 1/3, ..., 1/n for all n >= 1, >> then their total area is finite, although the sum of their radii is >> infinite. >> >> Can one fit these circles into a finite-sized rectangle? > > yes, they will fit inside a 1 x (11/6) rectangle for example. put each > circle inside a square, and then pack the squares like this: > http://www.stetson.edu/~efriedma/harmonic/ It is apparently still unknown whether one can pack squares of sides-lengths 1/1, 1/2, 1/3, etc. into a 1-by-(pi squared over six) rectangle. See Clive Tooth's web-page www.pisquaredoversix.force9.co.uk/Tiling.htm for an example of how one might get started. I asked Clive for background on this tiling, and he kindly forwarded to me copies of some relevant sci.math postings from last year (one by Clive himself and one by David Cantrell): ************************************************************************** From: "The Last Danish Pastry" <TheLastDanishPas...(a)yahoo.com> Newsgroups: sci.math Subject: Packing squares into a 1 x Zeta[2] rectangle Date: Sun, 14 Mar 2004 13:41:34 -0000 Since the sum of the reciprocals of the squares of the positive integers is pi^2/6, the question arises as to whether squares with sides 1, 1/2, 1/3, etc can be packed into a rectangle of size 1 by pi^2/6. A picture of such a packing appears at http://www.pisquaredoversix.force9.co.uk/Tiling.htm I know of no proof that such a packing is possible or impossible. A program which I wrote has recently packed the first one million such squares into such a rectangle. I wrote the program in Mathematica and it uses exact arithmetic throughout. The algorithm is simple: The program maintains a list of rectangles. Each member of this list is itself a list containing exactly two members: the short side of a rectangle followed by the long side of the rectangle. The program does not store the position or orientation of any rectangle. The list is stored in ascending short side order. Initially the list contains one entry: {1, pi^2/6}. The program inserts the squares in decreasing size order, starting with the 1x1 square. At each stage, to insert the square with side 1/n, the program finds the rectangle with the shortest short side which will accommodate the square. [In the case that there are two or more such rectangles with equal short sides, the program will pick the youngest one.] Suppose that this rectangle is a x b, where 1/n<=a<b. The program deletes this rectangle from the list and inserts two new rectangles (b-1/n) x a and (a-1/n) x 1/n into the list. - If any side of one of these new rectangles is zero, the rectangle is not inserted. - Each new rectangle is stored in {short side, long side} form. Thus, the list of rectangles grows by one per inserted square, except in the rare cases in which a rectangle of exactly the right width is found. A trivial optimisation, which took me a long time to notice, is that if the number of squares to be inserted is decided at the beginning, as N, say, then any rectangles narrower than 1/N can be discarded as soon as they appear. This has a dramatic effect on the memory requirements and running time of the program. In the case of N=10^6 the maximum length of the rectangle list is only 7,493 which occurs just after inserting the square with side 1/55,205. Thereafter the length of the list slowly decreases. After inserting the last square, with side 10^-6, there are just 7 rectangles in the list. Their dimensions are, roughly: 0.000001000264 x 0.000371508565 [1,000,000] 0.000001000289 x 0.000006561855 [ 999,736] 0.000001000332 x 0.000011950546 [ 999,710] 0.000001000424 x 0.000099911008 [ 999,668] 0.000001001705 x 0.000013366392 [ 999,575] 0.000001001771 x 0.000555285230 [ 998,297] 0.000621399972 x 0.000621402936 [ 998,232] The number in square brackets is the square insertion step on which the rectangle was created. Note that each rectangle, except the last, has a short side just exceeding 10^-6. The last one is huge in comparison to the others. Also, it is almost square, having an aspect ratio of about 1.00000477 . The exact size of that last rectangle may be expressed as: (pi^2/6 - ShortNumerator/ShortDenominator) by (LongNumerator/LongDenominator) Where ShortNumerator = 88148577597184641314023698687804198322622003067950 90277185502203375757868045576236778623499141581835 91095283100445795053089253938670874859361703421694 93589311823604254188777249571387340885945805010588 59158565482204748239750452173829861295626045336259 37611823762149119593932203343599960000593198902456 62151237628335467123582313067030461952701063806541 09667253352708582423851542412389977650553557102776 48867766805593783542952432276691372915179943637567 30921614687225494242879103415386639772227105415075 60643644416702161056703911777435451470868967261015 03285333687384626370723064421360623232628962182637 33785897530162588640506103091960735379749803817168 27337105343328553249474140253583305664128829433056 13496001065570446619534699711595669897241621797339 93388113729020710706312912651983440135427698863738 63455691221381048718834049233446561869585869886892 11066449759482338440176987851999633382542943944743 83052271691048660471857179765488027759900589960305 37378583137068870187987573969424873832345344118539 33368635075465276924658419905001027235726511505469 90353609271920928926254763071708700239940565565014 69168650106503612867362486635886531701170797485510 99375741498187773039269448405524049571692907463933 52292512770766034352574830725344567781634199947344 39128864912738319438009565992988317045941602472518 84296172734535520954249457165155850357846464984168 55299222999724423825677947429849212042460805178788 23214473791347512926915739195451188103089711423995 69891250713418835019305494967439261535807066257765 86799633983386818438909781178910531233498247023179 22489232680161587870238609995903489290522620579734 27913072409961919211050524749783993630732833119534 51221464170732847284504244076797981093088600479209 053071348915420741889 ShortDenominator = 53608160645434703140761211994317971700783074368336 23325975705041657962199994733358382994458318582660 77598233503372093023408990096331259105144227503324 99931830157705570560424971647310855286927378535483 51519419895212616165032657829920140448566211296935 08925318960134998789890442610040864611429495599467 55671385275409484022080990210675967224987014208598 03335236373663242927780013944648864149878730486125 39041613975393755631755935261748017063478349180906 68435262002209072815068783227098957825899390123056 87116154057047656878878907753207321889684985791605 16008019019744195226005265347943729214648741668979 90732236259531674523664568538104488968273268732774 74645971065348455713992347183400863276778739619943 89625588842652615699820699583835957017913827580111 39619239553717364061586421301662144411944880262864 13395969932087668263688142360819354641427777701996 11575083489100468980371536759296335301136691198138 39559100156995546068054947710363960181181289364408 47219039882502818687162480694856587042340603392188 93192097190331493671006154660882637398429894947140 66724834302759980279336075917538263518035481261929 09959571761983813160895335069812335369906345916646 57307263315174319702690110108769119880417800559541 00846379467033278426493445187132240090150811316237 11870955392306082427560405688265993364931620972062 42335013731582665984021398103261291575096750373611 78793366106368150944101270721191063173365978409196 32419016511795819355629216137822167151147383280072 17692953976640188298210375720290857531988737370969 01207985676688854219832216404101906724620701738590 65777947161752685898547108814129100143613045915161 23374032620084188029394058765229087587568694264890 44233796349785411756140522085279648064146173069741 478636801148014080000 LongNumerator = 86047691660315035868968439680618750650385496662337 69341618423224868557176707904056350747739644607152 19341301092244142985694427567208621603825424715808 50112490209730250303812592410205852634552809617406 54471220118615832220663814127702644186714156802514 89272173167125109946591599082250813475432136844437 68434150604827750301142594110541508119955806775639 48463821703879298880841441517861092382325647927555 17115984413353561811152746794319549286695056542231 75370181749913246697604501180515750468588947850814 96706860842334628047016110265861987131951269186913 55908484624794372227266755207694315503681976948427 59732973255837108405693520922709400140190847636749 60091554508265275883766083172672994896381234279724 35485615485280999872894822261731570552709443192476 07685741096374165452217373319152585412622236248087 75856424918727148070254116048744362672852782317829 99207549345144244124571110563211160343448412194752 79110755652853486307738886342254777707077572587992 29405091182393418319601369763056120887765700618802 27139753549245993550447831136020976532760818138519 98216552674127086312240961717818956038535226187300 43026597993187644795910836481706823332909020130603 09729716420449811214331443400182398534278484763571 27904525989212662382850630094162630962359551947273 97311922973034069310043168935737831265241199089047 46559558404697050977201086017144902884602248144108 71005578441670661613107026657746377160408235952122 70087369832576028781964223879066392055590320740847 64872057694761838719127407266757016871496087930551 58594303661605591550655979493954945142285417155556 97061447617516531128930865552526142619566789459050 16802730874016347300046435875463493515247956092289 80788952041557673741641666771939979618817508420910 11050430185421809507580791626321726210797393934779 33362836600904869583181 LongDenominator = 13847326202197605511604348541676281864027795818043 31494335574789145450920249049910013265993162156505 60831474662540178414423726822350848233301461067293 21368995360010230519749558542277057071473577106811 23548536664652068929533780687547853336157369555086 86435599910456774645119837370840414802536199558214 07527909448749766750406409444125057733978666254122 19353807926168688488715863115960118447262678159252 66974533884810419996115934738184899912380787401311 77856867483197701324817440576351765023155329523060 68640715686158929369832278778842403496825357945180 51999476544062632945516973292520225337991834072383 82846228280041100692245330583478338241014202245803 30323143236417275386997321382053499979484896390576 45426647084624716067636294896758072966317096343334 28169063530634898773783098553798071542939121086910 08006721058240228461124891602875033925435245224172 90909661066024272174911721070106393261965722571136 18608587173321485889447514408382277645342459166650 66976600733716756193252940889415454560334136529577 70915746535414564177067690689656587269581203051464 54295311030944948219808305598301111195229732416082 42817235468158091436079088616794858652804473035517 82817817155337872004025893330437425894039012377193 76447539452009335637045807403996380650616246802471 05691575600545667349590304669247493320414626833379 52612693846867728391114410877249463229225980473996 60123182871902607011087292407939606651288648107916 20645836880002779839862309998364888537462475216562 83515669632032306317521726424095159019732485843596 26389117354057755490411586373128011034072899097872 57238596906896860163242623358201011919041838989031 36072177994374094882330699298066695784371692394667 11940033790850742395135563324255474227834715815498 61014376098413906103604397447388393432391565219123 065068758213935671227536000 -- Clive Tooth http://www.clivetooth.dk ************************************************************************** Subject: Re: Packing squares into a 1 x Zeta[2] rectangle From: David W. Cantrell <DWCantr...(a)sigmaxi.org> Date: 18 Mar 2004 16:31:52 GMT Newsgroups: sci.math "The Last Danish Pastry" <TheLastDanishPas...(a)yahoo.com> wrote: > Since the sum of the reciprocals of the squares of the positive integers > is pi^2/6, the question arises as to whether squares with sides 1, 1/2, > 1/3, etc can be packed into a rectangle of size 1 by pi^2/6. > > A picture of such a packing appears at > http://www.pisquaredoversix.force9.co.uk/Tiling.htm > > I know of no proof that such a packing is possible or impossible. [snip] It would be good if people interested in this thread were aware of some of the history of this problem. To that end: _Unsolved Problems in Geometry_ by Croft, Falconer and Guy (Springer-Verlag, 1991) "D5. Packing unequal rectangles and squares in a square." (pp. 112-113) mentions a question similar to the one considered in this thread. D5 caused me, in 1996, to consider several packing problems of this sort. I found simple algorithms, which I implemented in Mathematica, which seemed to pack rectangles of decreasing size "perfectly". The algorithms worked for classic problems, such as packing the rectangles 1/n by 1/(n+1) into the unit square, as well as for interesting new packing problems. I did not publish my results then, hoping first to prove that the algorithms were adequate for packing all of the rectangles. In November 1999, spurred by Clive Tooth, I posted to sci.math a brief description of what I had done. See both parts of "Packing Rectangles of Decreasing Size" at <http://mathforum.org/discuss/sci.math/a/t/241167>. (Readers may also be interested in looking at the packing algorithm which Clive had used back then.) Unbeknownst to me in 1999, an algorithm for such packings had appeared previously in the literature: An Algorithm for Packing Squares, Marc M. Paulhus, _J. Combin. Theory Ser. A_ 82 (1997) 147-157. His algorithm is somewhat different from any of the algorithms which I had used (but I wonder if perhaps it is the same as Clive's current algorithm). For packing squares of side lengths 1/2, 1/3, 1/4,... into a rectangle 1/2 by 2(pi^2/6 - 1), Paulhus gives two rules: "Rule 1. Always place the next square in the corner of the smallest width rectangle into which it will fit." "Rule 2. After placing a square into the corner of a rectangle, always cut the remaining area into two rectangular pieces by cutting from the free corner of the square tot the longer side of the original rectangle." For older references, see those cited by Paulhus. The most recent articles known to me concerning such packings -- but again, I may not be up-to-date with the literature -- are the following: Perfect Square Packings, Adam Chalcraft, _J. Combin. Theory Ser. A_ 92 (2000) 158-172. Compactness Theorems for Geometric Packings, Greg Martin, <http://arxiv.org/PS_cache/math/pdf/0005/0005054.pdf>. Perfect Packings of Squares Using the Stack-Pack Strategy, Johan Wastlund, <www.mai.liu.se/~jowas/dcg.ps <http://www.mai.liu.se/~jowas/dcg.ps>>. Thoughtful comments are welcome. Regards, David W. Cantrell ************************************************************************** (Clive tells me that there's more discussion of this problem in sci.math, though I haven't checked.) Jim Propp

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[math-fun] a conjecture based on Euler pol. and the Fermat's little theorem.
by Simon Plouffe 08 Feb '06

08 Feb '06

hello everybody, I had this idea one night a while ago, if k = n^2 - n + 41 and if 2^(k-1) = 1 mod k then k is prime. It seems to work all the time, I verified for n up to 500,000,000. In other words : n^2 - n + 41 is never a 2-psp, a pseudo-prime in base 2 (that is the conjecture). There is another one too : if k = n^2 + 3 and if 2^(k-1) = 1 mod k then k is prime. I tried to construct an argument about the factorization of n^2 - n + 41 and 2-psp's but failed to see any convincing details. I have other values that seems to work also like k = n^2 + 163 (only 1 fail), another counter example is 103*n^2 - 3945*n + 34831 when n = 2400371 it does satisfy the little fermat theorem but the number is composite : 5315987*111635707. and that polynomial (103*n^2..) is known to produce a lot of primes. Based on this I launched a series of computations but (as you may know), it all comes back to the fact that the little fermat theorem produces primes very often and if we combine it with a simple g.f. like the Euler polynomial, chances are that it will produce many primes. Even by knowing this, the best probable prime I could find is about 35000 digits. 1) I could not find any reference about this. 2) So far n^2 + 3 and the Euler pol. always produces primes and avoid the 2-psp's elegantly. 3) Maybe there is alink between class numbers and the little fermat theorem but I can't see it. My question is do you see any connections ? Simon Plouffe

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[math-fun] Found in the IHP cellar! 2/2
by Christian Boyer 21 Nov '05

21 Nov '05

---- FAUQUEMBERGUE computations And I have the chance to have discovered, again in the cellar of the Institut Henri Poincaré, the original handwritten computations on Mersenne numbers M101, M103, and M109 as sent by Fauquembergue to André Gérardin. http://www.utm.edu/research/primes/mersenne/LukeMirror/lit/lit_008s.htm These computations done in 1913-1914 were... false, as found by Robinson 40 years later: http://www.utm.edu/research/primes/mersenne/LukeMirror/lit/lit_024s.htm Within the huge number of handwritten 0s and 1s, the game would be to locate the first line where the error occurs! I can send .JPG photos to interested people. Christian.

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Re: [math-fun] NYTimes: melting ice cubes cause water to rise ?
by dasimov＠earthlink.net 30 Sep '05

30 Sep '05

Henry Baker wrote that an old SAT question goes like this: << A cube of ice is floating in a glass of water that is filled to the top. (No discussion of meniscus.) When the ice melts, the water in the glass: A. overflows B. stays at the same level C. goes down D. none of the above >> (This sounds more like a question from a physics "achievement test" -- but be that as it may.) I would not say ice displaces the same "amount" of water, since "amount" could mean either mass or volume. The floating ice cube has the same weight as a) the water it displaces and b) the water it becomes when it melts. Thus a) and b) have the same volume. So when the cube melts, the water level shouldn't change. (So what's the business about floating ice raising water levels when it melts?) --Dan

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Re: [math-fun] NYTimes: melting ice cubes cause water to rise ?
by Henry Baker 30 Sep '05

30 Sep '05

You're correct about the potato. However, I was referring to the general technique, not the particular object. At 09:52 AM 9/30/2005, Mike Stay wrote: >On 9/30/05, Henry Baker <hbaker1(a)pipeline.com> wrote: >> As an aside, I recently saw a show on Italian cooking on cable TV, and the cook >> used the change in density of salt water to approximate the correct salinity for >> a salt broth. She put a potato into the water, and added enough salt until >> the potato started to float. That was the correct amount for her recipe. >> Something tells me that this technique was probably used for thousands of >> years. >Except that the potato is from the New World. Anyone know if Andean >indians use salt broths in their traditional cooking? >-- >Mike Stay >metaweta(a)gmail.com >http://math.ucr.edu/~mike

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[math-fun] Multiplicative squares
by Christian Boyer 30 Sep '05

30 Sep '05

Bonjour, As said yesterday, 302400 is the smallest possible product for 5x5 multiplicative squares. And 332640 is another new possible product smaller than our previous example with 362880. Here is one of the numerous possible examples with 302400: 12 35 1 40 18 36 2 24 7 25 14 45 15 4 8 5 16 42 30 3 10 6 20 9 28 coming from the family: aab cd 1 aaac abb aabb a aaab d cc ad bbc bc aa aaa c aaaa abd abc b ac ab aac bb aad magic product a^6 * b^3 * c^2 * d (example with a=2, b=3, c=5, d=7 for 302400, but of course the same family can be used for 423360, 475200, ...) Other numerous families are possible with the same magic product a^6 * b^3 * c^2 * d. Some words on the method used for nxn multiplicative squares. For each "good" P (means with several little factors), the program computes all the magic series of n distinct integers having the product P. For example with n=5: P=302400 has 3527 series, 332640 has 3661 series, 362880 has 3734 series. Then it analyzes combinations of n series looking if it is possible to get n series using n² distinct integers. If n series are found, they are the n rows: arrange them trying to get n columns and 2 diagonals. No great secret in the method... A specific page on multiplicative squares will be added in the next update of www.multimagie.com/indexengl.htm, planned beginning of October. Christian. ------------------------------------------ The 10 smallest magic products for 5x5 are: Prod. = 2^ * 3^ * 5^ * 7^ * 11^ *13^ 302400 6 3 2 1 0 0 << The SMALLEST 332640 5 3 1 1 1 0 362880 7 4 1 1 0 0 << Previously known 393120 5 3 1 1 0 1 << id 332640 403200 8 2 2 1 0 0 415800 3 3 2 1 1 0 423360 6 3 1 2 0 0 << id 302400 443520 7 2 1 1 1 0 475200 6 3 2 0 1 0 << id 302400 491400 3 3 2 1 0 1 << id 415800 ------------------------------------------

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Re: [math-fun] NYTimes: melting ice cubes cause water torise ?
by dasimov＠earthlink.net 30 Sep '05

30 Sep '05

Dirk writes: << If you consider the spherical shape of the earth and the gravity not pointing in the same direction everywhere, I think it will be different (but I did no calculations). In the extreme case, when all the surface of a water planet is covered by ice, the ice will not sink at all because the gravity cancels out everywhere. When this ice block melts, the water level will rise... >> Surely there is more involved in this spherical ice shell than mere floating! (Imagine a spherical shell of radius ~1 meter, with thickness = 1/10 meter, made of iron, and now cut it into, say, 4 spherically-curved 120-degree triangular slabs, leaving them in position. Now imagine trying to push all 4 pieces towards the center at the same time. They will get in each others' way.) If instead we chop up the spherical shell of ice into pieces -- and get rid of some ice so that [the cones on the pieces from the earth's center] are mutually disjoint -- then starting from this situation instead of from the complete spherical shell of ice -- then when *this* ice all melts, the water level should not change perceptibly. I don't see any gravity cancellation here, or in the original situation. --Dan

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[math-fun] another kind of median (trying again)
by James Propp 30 Sep '05

30 Sep '05

The picture was supposed to show a rectangle of width 1 and height 2 whose bottom is centered at x=1, and to the right of it, a rectangle of width 1 and height 1 whose bottom is centered at x=2. The base of the first rectangle goes from x=1/2 to x=3/2, and the base of the second rectangle goes from x=3/2 to x=5/2. The total area under the histogram is (1)(2)+(1)(1) = 3. The area to the left of the line x=5/4 is (5/4-1/2)(2) = 3/2, which is half of the total area. So x=5/4 is the "median". Jim

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Antwort: Re: [math-fun] NYTimes: melting ice cubes cause water to rise ?
by dirk.lattermann＠bgs-ag.de 30 Sep '05

30 Sep '05

If you consider the spherical shape of the earth and the gravity not pointing in the same direction everywhere, I think it will be different (but I did no calculations). In the extreme case, when all the surface of a water planet is covered by ice, the ice will not sink at all because the gravity cancels out everywhere. When this ice block melts, the water level will rise... Dirk Henry Baker <hbaker1(a)pipeline.com>@mailman.xmission.com am 30.09.2005 05:52:51 Bitte antworten an math-fun <math-fun(a)mailman.xmission.com> Gesendet von: math-fun-bounces+dirk.lattermann=bgs-ag.de(a)mailman.xmission.com An: dasimov(a)earthlink.net, math-fun <math-fun(a)mailman.xmission.com> Kopie: Thema: Re: [math-fun] NYTimes: melting ice cubes cause water to rise ? That was the point of my original posting -- the NYTimes thought that melting floating ice would raise the water level -- they were obviously wrong. At 04:09 PM 9/29/2005, dasimov(a)earthlink.net, math-fun wrote: >(So what's the business about floating ice raising water levels when it melts?) > >--Dan _______________________________________________ math-fun mailing list math-fun(a)mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun

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[math-fun] NYTimes: melting ice cubes cause water to rise ?
by Henry Baker 29 Sep '05

29 Sep '05

Let me see -- melting floating ice makes the water level rise? Isn't this a standard SAT question? I guess Mr Revkin's verbal SAT scores exceed his math SAT scores. ------------------ The New York Times September 29, 2005 In a Melting Trend, Less Arctic Ice to Go Around By ANDREW C. REVKIN The floating cap of sea ice on the Arctic Ocean shrank this summer to what is probably its smallest size in at least a century of record keeping, continuing a trend toward less summer ice, a team of climate experts reported yesterday. That shift is hard to explain without attributing it in part to human-caused global warming, the team's members and other experts on the region said. The change also appears to be headed toward becoming self-sustaining: the increased open water absorbs solar energy that would otherwise be reflected back into space by bright white ice, said Ted A. Scambos, a scientist at the National Snow and Ice Data Center in Boulder, Colo., which compiled the data along with the National Aeronautics and Space Administration. "Feedbacks in the system are starting to take hold," Dr. Scambos said. The data was released on the center's Web site, www.nsidc.org. The findings are consistent with recent computer simulations showing that a buildup of smokestack and tailpipe emissions of greenhouse gases could lead to a profoundly transformed Arctic later this century, when much of the once ice-locked ocean would routinely become open water in summers. Expanding areas of open water in the summer could be a boon to whales and cod stocks, and the ice retreat could create summertime shipping shortcuts between the Atlantic and the Pacific. But a host of troubles lie ahead as well. One of the most important consequences of Arctic warming will be increased flows of meltwater and icebergs from glaciers and ice sheets, and thus an accelerated >>rise in sea levels,<< threatening coastal areas. The loss of sea ice could also hurt both polar bears and Eskimo seal hunters. The Arctic ice cap always grows in the winter and shrinks in the summer. The average minimum area from 1979, when precise satellite mapping began, until 2000 was 2.69 million square miles, similar in size to the contiguous area of the United States. The new summer low, measured on Sept. 19, was 20 percent below that. Before 1979, scientists estimated the size of the ice cap based on reports from ships and airplanes. The difference between the average ice area and the area that persisted this summer was about 500,000 square miles, an area about twice the size of Texas, the scientists said. This summer was the fourth in a row with the ice cap areas sharply below the long-term average, said Mark C. Serreze, a senior scientist at the snow and ice center and a professor at the University of Colorado, Boulder. Dr. Scambos said the consecutive reductions in the ice cap "make it pretty certain a long-term decline is under way." A natural cycle in the polar atmosphere called the Arctic oscillation, which contributed to the reduction in Arctic ice in the past, did not appear to be a factor in the past several years, Dr. Serreze said. He said the role of accumulating greenhouse gas emissions had become increasingly apparent with rising air and sea temperatures. Still, many scientists say it is not yet possible to determine what portion of Arctic change is being caused by rising levels of carbon dioxide and other emissions from human sources and how much is just climate's usual wiggles. Dr. Serreze and other scientists said that more variability could lie ahead and that the area of sea ice could actually increase some years. But the scientists have found few hints that other factors, like more Arctic cloudiness in a warming world, will reverse the trend. "With all that dark open water, you start to see an increase in Arctic Ocean heat storage," Dr. Serreze said. "Come autumn and winter that makes it a lot harder to grow ice, and the next spring you're left with less and thinner ice. And it's easier to lose even more the next year." The result, he said, is that the Arctic is "becoming a profoundly different place than we grew up thinking about." Other experts on Arctic ice and climate disagreed on details. For example, Ignatius G. Rigor at the University of Washington said the change was probably linked to a mix of factors, including influences of the atmospheric cycle. But he agreed with Dr. Serreze that the influence from greenhouse gases had to be involved. "The global warming idea has to be a good part of the story," Dr. Rigor said. "I think we have a different climate state in the Arctic now. All of these feedbacks are starting to kick in and really snuffing the ice out by the end of summer." Other experts expressed some caution. Claire L. Parkinson, a sea ice expert at NASA's Goddard Space Flight Center in Greenbelt, Md., said a host of changes in the Arctic - including rising temperatures, melting permafrost and shrinking sea ice - were consistent with human-caused warming. But she emphasized that the complicated system was still far from completely understood. William L. Chapman, a sea ice researcher at the University of Illinois Urbana-Champaign, said it was important to keep in mind that the size of the ice cap could vary tremendously, in part because of changes in wind patterns, which can cause the ice to heap up against one Arctic shore or drift away from another.

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