Знание без граници Le Trou Noir
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- ESPACES À COURBURE VARIABLE : _____________________________________________________________________ SPACES WITH VARIABLE CURVATURE
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| Знание без граници
Le Trou Noir The Black Hole Черната дупка Jean-Pierre Petit Jean-Pierre Petit Жан-Пиер Пети
Translated by : Превод : Елена Ст. Владова (Elena Vladova) _____________________________________________________________________ 1 Si cet Univers est le meilleur des univers possibles, que sont donc les autres ? _____________________________________________________________________ If this universe is the best of all possible worlds, I’m glad I don’t live in any of the others. _____________________________________________________________________
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3 Monsieur Albert ! Qu'est-ce que vous faites ? _____________________________________________________________________ MR. Albert ! What are you doing ? _____________________________________________________________________ Господин Албер, какво правите? __________________________________________________________________ Tiens, Anselme, assieds-toi _____________________________________________________________________ Haf’ a seat, Archibald. _____________________________________________________________________ Чакай, Анселм. Седни. _____________________________________________________________________ C'est quoi ? _____________________________________________________________________ What’s this ? _____________________________________________________________________
_____________________________________________________________ Le Cosmol, Anselme, le Cosmol ...... .... qui ouvre la Porte _____________________________________________________________________ Cosmol, Archibald : der Cosmol ... Dat points der way.
________________________________________________________________________ La Porte, quelle Porte ? Je ne comprends pas _____________________________________________________________________ The way ? What way ? I don’t understand. _____________________________________________________________________ Вратата? Каква врата? Не разбирам. ________________________________________________________________________ Comprendre ! Ah oui ... ... comprendre ... _____________________________________________________________________ Understand ? Oh, Ja, unnerstan’ ... _____________________________________________________________________ Да разбереш? О, да... ... да разбереш... _____________________________________________________________________ Les autres disent : démarche logique ... c'est de la foutaise ! _____________________________________________________________________ Everyone says you haf’ to think logically ... dat’s garbage ! _____________________________________________________________________ Хората го наричат логическо мислене. ... това са глупости. _____________________________________________________________________ Seul ... le Cosmol ... .. ouvre la Porte .. _____________________________________________________________________ Only ... Cosmol ... can point der way ...
Само Космолът може да отвори Вратата. _______________________________________________________________ ... Foutaise ... _____________________________________________________________________ ... Garbage ... _____________________________________________________________________ Дрън-дрън... La Porte, oui la Porte ... _____________________________________________________________________ The way ? Oh, that way ! way out ...
Вратата, да Вратата... __________________________________________________________ 4 Du Grand Univers _____________________________________________________________________ .......... _____________________________________________________________________
_____________________________________________________________________ ... Univers ... _____________________________________________________________________ .......... _____________________________________________________________________ Вселена...
... Univers ... _____________________________________________________________________ .......... _____________________________________________________________________
Cosmol
Космол _______________________________________________________ Monsieur Albert ! _____________________________________________________________________ MR. Albert ?
Господин Албер! _________________________________________ Sophie !
_____________________________________________________________________ Sophie ?
_____________________________________________________________________ Софи!
Hum ... _____________________________________________________________________ Hum ... Х-м...
_____________________________________________________________________ Encore ces nuées ... _____________________________________________________________________ These cloups again ... _____________________________________________________________________ Пак има облаци... __________________________________________________________ ........
5 Once more, Archie sets out to explore The world of Mists. _____________________________________________________________________ Une fois de plus, Anselme part explorer des mondes nébuleux. Анселм отново се кани да изследва мъгляви светове...
ou une sorte de rouleau à peinture. _____________________________________________________________________ Gosh, what’s this gadget ? It looks like a roller for a tennis court ... or maybe for painting things.
Я виж, какво е това? Прилича на валяк за тенис корт или пък за боядисване. _____________________________________________________________________ On peut le déplacer TOUT DROIT, sans effort. Par contre, impossible de dévier d'un poil à DROITE ou à GAUCHE. _____________________________________________________________________ I can push it STRAIGHT AHEAD without any trouble. But it won’t budge a hair’s breadth to RIGHT or LEFT. _____________________________________________________________________
__________________________________________________________ A quoi sert cette poignée ? Tiens, elle supprime l'adhérence et me permet de temps en temps de changer de direction. _____________________________________________________________________ I wonder what this handle’s for ? Aha ! It unsticks the roller and lets me change direction every so often. _____________________________________________________________________ За какво служи тази дръжка? Я виж, тя раздвижва валяка и ми позволява от време на време да сменя посоката. __________________________________________________________ Grâce à cet appareil, Anselme peut tracer les GÉODÉSIQUES d'une surface. A l'aide de trois géodésiques, Anselme peut tracer des TRIANGLES : _____________________________________________________________________ With the aid of this strange device, Archie can draw GEODESICS on a surface. Using these geodesics, he can draw TRIANGLES : _____________________________________________________________________ Благодарение на този апарат Анселм може да начертае ГЕОДЕЗИЧЕСКИТЕ ЛИНИИ върху една повърхност. Благодарение на трите геодезически линии Анселм може да начертае ТРИЪГЪЛНИЦИ. ___________________________________________________________________ Une surface est un ESPACE À DEUX DIMENSIONS. C’est-à-dire qu’il faut DEUX QUANTITÉS pour y repérer la position d’un point, deux coordonnées. _____________________________________________________________________ A surface is a 2 - DIMENSIONAL SPACE. That is, you need TWO numbers – two coordinates – to specify the position of a point.
___________________________________________________________________ 6 Voyons, quand l'espace est EUCLIDIEN, la somme des angles de mon triangle vaut 180 °. _____________________________________________________________________ Lesse ... If the space is EUCLIDEAN, then the sum of the angles of a triangle is 180°. * _____________________________________________________________________ Да видим. Когато пространството е ЕВКЛИДОВО, сумата от ъглите е 180 °.(*) _____________________________________________________________________ Quand l'espace a une courbure négative, cette somme est INFÉRIEURE à 180 degrés _____________________________________________________________________ When the space has negative curvature, the sum is LESS than 180°.
Когато пространството има отрицателна кривина, сборът е ПО-МАЛЪК от 180°. _____________________________________________________________________ Dans un espace à courbure POSITIVE, la somme est SUPÉRIEURE à 180 degrés _____________________________________________________________________ In a space of positive curvature, it’s GREATER than 180°. _____________________________________________________________________ В пространство с ПОЛОЖИТЕЛНА кривина сумата е ПО-ГОЛЯМА от 180°. _____________________________________________________________________ ESPACES À COURBURE VARIABLE : _____________________________________________________________________ SPACES WITH VARIABLE CURVATURE _____________________________________________________________________ ПРОСТРАНСТВА С ПРОМЕНЛИВА КРИВИНА _____________________________________________________________________ J'ai inventé un curvimètre. Il est constitué de trois lamelles élastiques pouvant tourner librement autour de trois rivets A, B, C. _____________________________________________________________________ Here’s the cunning inventor with his new brainchild, the CURVIMETER. It’s made from three flexible strips which can turn freely about three rivets A, B, and C. _____________________________________________________________________ Изобретих уред за измерване на кривините. Той се състои от три еластични пластини, които могат да се въртят свободно около трите нита A, B и C. _____________________________________________________________________ Il suffit de le plaquer sur une surface et de mesurer les angles à l'aide des trois rapporteurs pour connaître la COURBURE LOCALE. _____________________________________________________________________ To find the LOCAL CURVATURE it suffices to lay the curvimeter on the surface and measure the angles with the built-in protractors.
COURBURE POSITIVE _____________________________________________________________________ POSITIVE CURVATURE _____________________________________________________________________ ПОЗИТИВНА КРИВИНА _____________________________________________________________________ COURBURE NÉGATIVE _____________________________________________________________________ NEGATIVE CURVATURE _____________________________________________________________________ НЕГАТИВНА КРИВИНА
(*) Pour plus de détails, voir le GÉOMÉTRICON, du même auteur, Editions BELIN. _____________________________________________________________________ (*) For further details, see here’s looking at euclid in the same series. _____________________________________________________________________ (*) За повече подробности, виж ГЕОМЕТРИКОН от същия автор. _____________________________________________________________________ 7 Cette bosse ménagée dans un plan est constituée d’une région centrale à courbure positive, entourée par une région à courbure négative. _____________________________________________________________________ This bump in the plane is made from a central region of positive curvature, surrounded by a region of negative curvature. _____________________________________________________________________ Тази изпъкналост в равнинаната се състои от централна част с положителна кривина, заобиколена от зона с отрицателна кривина. _____________________________________________________________________ ZÔNE À COURBURE POSITIVE ZÔNE À COURBURE NÉGATIVE ZÔNE À COURBURE NULLE _____________________________________________________________________ REGION OF POSITIVE CURVATURE REGION OF NEGATIVE CURVATURE REGION OF ZERO CURVATURE _____________________________________________________________________ ЗОНА С ПОЛОЖИТЕЛНА КРИВИНА ЗОНА С ОТРИЦАТЕЛНА КРИВИНА ЗОНА С НУЛЕВА КРИВИНА _____________________________________________________________________ Du point de vue de la COURBURE, le CREUX est identique à la BOSSE _____________________________________________________________________ From the point of view of curvature, a dent is just like a bump. _____________________________________________________________________ От гледна точка на КРИВИНАТА ВДЛЪБНАТИНАТА е равна на ИЗПЪКНАЛОСТТА. _____________________________________________________________________
COURBURE NULLE COURBURE NÉGATIVE _____________________________________________________________________ POSITIVE CURVATURE ZERO CURVATURE NEGATIVE CURVATURE _____________________________________________________________________ ПОЛОЖИТЕЛНА КРИВИНА НУЛЕВА КРИВИНА ОТРИЦАТЕЛНА КРИВИНА _____________________________________________________________________ Sauf erreur, ceci est un TORE. _____________________________________________________________________ Crikey, if I ain’t mistaken, h’it’s a blinkin’ TORUS. _____________________________________________________________________ Ако не греша, това е геометрична фигура тор. _____________________________________________________________________ Oui, il y a une bande à courbure positive, une autre à courbure négative, séparées par une frontière où la courbure est nulle. _____________________________________________________________________ I’m a VIRGO myself ... Oh, I see. Yes, there’s a band of positive curvature and a band of negative curvature, separated by a frontier having zero curvature. _____________________________________________________________________ Да, една зона е с положителна кривина, друга с отрицателна кривина, а двете са разделени от граница с нулева кривина. _____________________________________________________________________ Cette dernière pouvant être déterminée en mettant le tore en sandwich entre deux plans.
To find the frontier, make a TORUSBUGER using two planes for buns. _____________________________________________________________________ Тази граница може да бъде изчислена и да превърне тора на сандвич между две равнини. _____________________________________________________________________ Mon cher Tirésias, avez-vous réalisé que votre coquille était un espace bidimensionnel à courbure variable ? _____________________________________________________________________ Tiresias, old fruit, ‘as it ever struck yer that shell is just a two-dimensional space wiv variable curvyture ? _____________________________________________________________________ Уважаеми Тирезий, забелязахте ли, че вашата черупка е двуизмерно пространство с променлива кривина? _____________________________________________________________________ mi !..
_____________________________________________________________________ eeep !
Ами-и-и... _____________________________________________________________________ Léon, laisse Tirésias tranquille !
Lenny, stop bugging Tiresias ! _____________________________________________________________________ Леон, остави Тирезий на мира! _____________________________________________________________________ 8
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_____________________________________________________________________ Tu vas voir, Anselme, il y a des choses plus étranges encore. _____________________________________________________________________ You’ll soon see, Archie, that there are even stranger things. _____________________________________________________________________ Анселм, ще видиш, че има и още по-странни неща. _____________________________________________________________________ Dépêche-toi, Tirésias, j’ai soif de connaître ...
Buzz off, Tiresias – I sense a thirst for knowledge coming upon me ... _____________________________________________________________________ Побързай, Тирезий, искам да науча още... _____________________________________________________________________ Attends-moi ! _____________________________________________________________________ Wait for boo-boo ! _____________________________________________________________________ Чакай ме! _____________________________________________________________________ Tu vois, Tirésias, je vais MAILLER ma surface en entrecroisant des géodésiques, ce qui me donnera des tas de triangles _____________________________________________________________________ Watch, Tiresias – I’m going to draw a grid on the surface by meshing geodesics. That’ll give me HEAPS of triangles. _____________________________________________________________________ Виждаш ли, Тирезий, сега ще сплета в МРЕЖА моето пространство и като пресека геодезическите линии, ще получа много триъгълници. _____________________________________________________________________ Coquille à courbure variable ... Je t’en ficherais, moi !!.. _____________________________________________________________________ Typical – Archie’s meshing about again ! _____________________________________________________________________ Черупка с променлива кривина! _____________________________________________________________________ Alors là, je n’y comprends plus rien ! Que se passe-t-il autour de ce point P ? _____________________________________________________________________ Oh lord, I can’t see what’s going on at all ! What’s happening around this point P ? _____________________________________________________________________ Нищо не разбирам. Какво става около тази точка P ? _____________________________________________________________________ Tu n’as qu’à utiliser ton curvimètre. _____________________________________________________________________ The cunning inventor has forgotten to use his curvimeter. _____________________________________________________________________ Трябва да използваш уреда за измерване на кривини. _____________________________________________________________________ 9 Enfin, Sophie, que se passe-t-il ? Si le triangle du curvimètre ne contient pas ce point P, il indique une courbure nulle. _____________________________________________________________________ Hey, Sophie – What’s going on ? If the curvimeter’s triangle doesn’t contain the point P, it says the curvature is zero. _____________________________________________________________________ Софи, какво става? Ако триъгълникът на уреда за измерване на кривини не съдържа тази точка Р, той показва нулева кривина. _____________________________________________________________________ Mais si le point P est dans le triangle, alors c’est courbe ! _____________________________________________________________________ But if P is inside the triangle, then the space is curved ! _____________________________________________________________________ Но точка Р е в самия триъгълник, значи пространството е изкривено. _____________________________________________________________________ C’est un point cônique. Tiens, regarde, je prends un plan, J’ENLÈVE un secteur d’angle θ et je recouds. _____________________________________________________________________ It’s a CONICAL POINT. Look – suppose I take a plane, cut out a sector with angle θ, and join it up again. _____________________________________________________________________ Това е ТОЧКА НА КОНУСА. Виж, взимам една плоскост, отрязвам един сектор с ъгъл θ и отново зашивам плоскостта. _____________________________________________________________________ PLAN
PLANE
ПЛОСКОСТ
_____________________________________________________________________ J’obtiens un cône que nous appellerons un POSICÔNE _____________________________________________________________________ I get a kind of cone, which we’ll call a POSICONE _____________________________________________________________________ Получавам конус, който наричаме ПОЛОЖИТЕЛЕН КОНУС. _____________________________________________________________________ Vous pouvez vérifier, avec du carton. Un rouleau de papier collant vous aidera à matérialiser facilement les géodésiques. _____________________________________________________________________ You can test it out with a piece of cardboard. A roll of sticky tape will help you find geodesics easily. _____________________________________________________________________ Можете да направите същото с картон. Ролка тиксо ще ви помогне да намерите геодезическите линии. _____________________________________________________________________ 10 Bon, alors, si mon triangle contient le sommet d’un cône, la somme de ses angles est toujours supérieure à 180° ! _____________________________________________________________________ Oh, I see ! If my triangle contains the tip of a cone then the sum of the angles is always greater than 180°. _____________________________________________________________________ Добре. Значи, ако моят триъгълник съдържа върха на конус, сборът от ъглите е винаги по-голям от 180° ! _____________________________________________________________________ Pas si vite ! En entaillant mon plan je vais maintenant au contraire RAJOUTER un secteur d’angle θ. _____________________________________________________________________ Hold your horses ! It’s not that simple ! Suppose instead I slit the plane and add a sector with angle θ.
Не бързай. Когато сглобявам моята плоскост, ще направя обратното – ще ДОБАВЯ един сектор с ъгъл θ. _____________________________________________________________________ Alors ... Ça fera un NÉGACÔNE ? _____________________________________________________________________ Don’ tell me – yer gets a blinkin’ NEGACONE. _____________________________________________________________________ Значи, ще се получи ОТРИЦАТЕЛЕН КОНУС. _____________________________________________________________________ Cette fois-ci, quand le triangle entoure le point P, la somme des angles vaut 180° - θ ! _____________________________________________________________________ This time, if the triangle contains the point P, the angle-sum will be 180°- θ, smaller than 180°. _____________________________________________________________________ Този път, когато точка Р е в триъгълника, сумата от ъглите ще бъде 180° - θ ! ___________________________________________________________________ Mais, de nouveau, quand le point est en dehors du triangle, la somme vaut 180°. _____________________________________________________________________ But if the point is outside the triangle, the sum is 180° again. _____________________________________________________________________ Но когато точката е извън триъгълника, сумата е отново 180°. _____________________________________________________________________ Cette propriété des cônes est indépendante de la taille du triangle, que celui-ci soit minuscule ou gigantesque.
This property of cones is independent of the size of the triangle : you get the same result whether it’s large or small. _____________________________________________________________________ Това свойство на конусите не зависи от размера на триъгълника, дали той е малък или голям. _____________________________________________________________________ 11 Mais ... quand même ! ... c’est courbe ou c’est pas courbe ? _____________________________________________________________________ But – That’s crazy ! Is the stupid thing curved, or not ? _____________________________________________________________________ Но, все пак... Това кривина ли е или не? _____________________________________________________________________ Un point cônique, Anselme, c’est de la courbure concentrée _____________________________________________________________________ A conical point, Archie, is a concentrated piece of curvature. _____________________________________________________________________ Точката на конуса, Анселм, е концентрична кривина. _____________________________________________________________________ Entre les points côniques, l’espace est euclidien, sans courbure
Away from the conical point, the space is (locally) euclidean, without any curvature. _____________________________________________________________________ Пространството между точките на конуса е евклидово, без кривина. _____________________________________________________________________ L’angle θ est la mesure de cette quantité de courbure _____________________________________________________________________ The angle θ measures the amount of curvature at the conical point. _____________________________________________________________________ Ъгъл θ е мярката за тази кривина. _____________________________________________________________________ Ouvre ton cône et mets le à plat _____________________________________________________________________ Open your cone and lay it out flat. _____________________________________________________________________ Разтвори твоя конус и го разгъни. _____________________________________________________________________ POSICÔNE
_____________________________________________________________________ POSICONE _____________________________________________________________________ ПОЛОЖИТЕЛЕН КОНУС _____________________________________________________________________ Voici le résultat de l’opération, faite par Anselme, dans le cas d’un cône à courbure positive. _____________________________________________________________________ Here’s what Archie got, when he opened out a cone having positive curvature.
Ето резултата от операцията, извършена от Анселм, когато конусът е с положителна кривина. _____________________________________________________________________ Et dans le cas d’un cône à courbure négative _____________________________________________________________________ Where as, in the case of negative curvature ... _____________________________________________________________________ И когато конусът е с отрицателна кривина. _____________________________________________________________________ NÉGACÔNE
NEGACONE _____________________________________________________________________ ОТРИЦАТЕЛЕН КОНУС _____________________________________________________________________ 12 Prenons une surface PLANE et maillons-la avec des géodésiques formant un quadrillage régulier. On dira qu’on a PAVÉ cette surface avec des carrés, tous identiques. Si nous suivons une TRAJECTOIRE, un TRAJET, tel qu’on coupe les côtés des carrés successifs sous un même angle, ce trajet s effectuera selon une géodésique de la surface. La Direction _____________________________________________________________________ Take a PLANE surface and rule it with geodesics to from a regular square grid. We say that the surface has been tiled with identical squares. If we follow a path, or TRAJECTORY, that cuts each successive square at the same angle, then this path will always be a geodesic on the surface. The Boss _____________________________________________________________________ Да вземем едно РАВНО пространство и да прокараме през него геодезическите линии, за да получим решетка от правилни квадрати. Все едно сме НАЧЕРТАЛИ в пространството еднакви квадрати. Ако прокараме ТРАЕКТОРИЯ така, че да отрежем краищата на последователни квадрати под един и същи ъгъл, то тази траектория ще мине по геодезическата линия на повърхността. Ръководството _____________________________________________________________________ Mais, pourquoi ne pas faire cela sur une sphère ? _____________________________________________________________________ So what stops me from doing the same on a sphere ? _____________________________________________________________________ Но защо това да не може да се направи и с една сфера? _____________________________________________________________________ Primo, essaie de PAVER une sphère avec des carrés, bien jointifs, tu m’en diras des nouvelles. _____________________________________________________________________ Well, you might start by trying to tile a sphere neatly with squares that would be novel.
Ако успееш да НАЧЕРТАЕШ в сферата последователни квадрати, ела да ми кажеш! _____________________________________________________________________ Les méridiens d’une sphère sont des géodésiques de celle-ci. Un trajet coupant ces méridiens sous un angle constant, différent de 90°, mènerait invariablement vers l’un des PÔLES !
_____________________________________________________________________ The meridians of a sphere are its geodesics. A path cutting the meridians at a constant angle, other than 90°, will invariably wind towards one of the POLES ! _____________________________________________________________________ Меридианите на една сфера са геодезическите й линии. Линия, която преминава през тези меридиани под постоянен ъгъл, различен от 90°, ще ни отведе неминуемо до ЕДИН ОТ ПОЛЮСИТЕ. _____________________________________________________________________ Une navigation à cap constant mène ... au pôle ! _____________________________________________________________________ Navigation on a fixed heading leads to ... the pole ! _____________________________________________________________________ Плаването с постоянен курс води до... полюса. _____________________________________________________________________ PÔLE
POLE
ПОЛЮС
_____________________________________________________________________ 13 En coupant les méridiens de la sphère à 90°, je me déplacerais suivant des parallèles _____________________________________________________________________ If I cuts the meriddyuns of a sphere at 90°, I go along one o’ the parallels. _____________________________________________________________________ Пресичайки меридианите на сферата на 90°, ще се движа по паралелите. _____________________________________________________________________ parallèles qui ne sont pas de géodésiques. Vu ! (*) _____________________________________________________________________ But parallels ain’t geo-deesicks. Gottit ! (*) _____________________________________________________________________ Паралели, които не са геодезически линии ! (*) ____________________________________________________________________ Je peux recouvrir une surface plane, euclidienne, à l’aide d’éléments plans, quadrillés _____________________________________________________________________ I can get a plane euclidean surface by fitting together a lot of flat grids. _____________________________________________________________________ Мога отново да получа равно евклидово пространство с помощта на квадратни елементи. _____________________________________________________________________ Et si je me déplace en coupant ces maillages sous un angle constant, à condition de bien assurer les raccords, de proche en proche, j’obtiendrai une géodésique. _____________________________________________________________________ And if I move by cutting the grid lines at a fixed angle, remembering to keep everything consistent when I switch grids, then I’ll get a geodesic. _____________________________________________________________________
_____________________________________________________________________ POSICÔNE _____________________________________________________________________ POSICONE ____________________________________________________________________ ПОЛОЖИТЕЛЕН КОНУС _____________________________________________________________________ NÉGACÔNE
NEGACONE _____________________________________________________________________ ОТРИЦАТЕЛЕН КОНУС _____________________________________________________________________ Ce moyen simple donne également les géodésiques du cylindre, qui sont en forme de ressort à boudin
You can do the same thing on a cylinder – just like a salami in a string bag. _____________________________________________________________________ По този прост начин се получават и геодезическите линии на цилиндъра, които имат формата на спирална пружина. _____________________________________________________________________ (*) On ne peut les tracer, sur la sphère, à l’aide de ruban adhésif (sauf l’équateur). _____________________________________________________________________ (*) You can’t draw a parallel on a sphere with sticky tape, except at the equator. Try it on a basketball. _____________________________________________________________________ (*) Не можем да очертаем паралелите върху сферата с помощта на тиксо, с изключение на Екватора. _____________________________________________________________________ 14 Voici pourquoi la somme des angles d’un triangle, sur un posicône, s’accroît de l’angle de découpe θ : _____________________________________________________________________ Here’s why the angle-sum of a triangle, on a posicone, is increased by the angle θ of the cut. _____________________________________________________________________ Ето защо сумата от ъглите на един триъгълник на позитивен конус се увеличава от ъгъла на рязане θ. _____________________________________________________________________ Anselme va maintenant construire des cônes particuliers, dans lesquels la régularité du maillage peut être conservée La Direction _____________________________________________________________________ Archibald higgins will now construct special cones, which keep the tiling regular. The Boss _____________________________________________________________________ Сега Анселм ще направи отделни конуси, с правилно разграфяване. Ръководството _____________________________________________________________________ Là, j’enlève 90°. _____________________________________________________________________ There, I’ve cut out 90°. _____________________________________________________________________ Ето, отрязвам под 90°. _____________________________________________________________________ Sur un tel cône, tu peux tracer des triangles rectangles équilatéraux.
On this cone you can draw right-angled equilateral triangles. _____________________________________________________________________ Върху този конус можеш да нарисуваш правоъгълни равностранни триъгълници. _____________________________________________________________________ 15 Maintenant, j’enlève un secteur de 180°. _____________________________________________________________________ Now I excise a sector of 180°. _____________________________________________________________________ Сега ще отрежа сектор под 180°. _____________________________________________________________________ Sur un tel cône, la somme des angles d’un triangle vaut 360° _____________________________________________________________________ On this cone, yer gits a h’angle – sum of 360°. _____________________________________________________________________ Върху такъв конус сумата от ъглите на един триъгълник ще е равна на 360°. _____________________________________________________________________ Ce qui signifie que l’on pourrait tracer dessus, à l’aide de ses géodésiques, un triangle ayant trois angles égaux à 120°, donc obtus.
Which implies that you can draw on it, using geodesics, a triangle having three angles of 120° - that is, OBTUSE. _____________________________________________________________________ Което означава, че отгоре може да се начертае с помощта на геодезическите линии триъгълик с три ъгъла по 120°, тоест ТЪПИ. _____________________________________________________________________ Et il se refermerait quand même ? _____________________________________________________________________ And it still closes up ? That’s weird. _____________________________________________________________________ Дори и ако се затвори? _____________________________________________________________________ Hum ... ____________________________________________________________________ Hum ... _____________________________________________________________________ Х-м... _____________________________________________________________________ Bien sûr, mon cher Tirésias, c’est vous qui êtes obtus ! _____________________________________________________________________ Seems ter me, Tiresias, that YOU’RE the one wot’s h’obtuse ! _____________________________________________________________________ Разбира се, уважаеми Тирезий, вие сте тъп! _____________________________________________________________________ mi !
Me ?
Аз? _____________________________________________________________________ 16 Sur ce cône, on peut tracer des BIANGLES, la somme des angles valant 180°. _____________________________________________________________________ On this cone you can draw BIANGLES. The angle-sum is 180°. _____________________________________________________________________ Върху този конус могат да се начертаят ДВУЪГЪЛНИЦИ, сумата от чиито ъгли ще е равна на 180°. _____________________________________________________________________ Attendez ! là, je ne comprends plus ... On parlait des triangles. Voilà maintenant des BIANGLES. Pourquoi pas, la prochaine fois, des .... monoangles ?!?!.... _____________________________________________________________________ 'Old it ! Nar yer lorst me ... First it wuz triangles. Nar it’s blinkin’ biangles. Wozzit gonna be next ? monoangles ?
Чакайте! Вече нищо не разбирам! Ставаше въпрос за триъгълници. Сега пък за ДВУЪГЪЛНИЦИ. А другия път за едноъгълници ли ще говорим? _____________________________________________________________________ LE CÔNE VU DE DESSUS _____________________________________________________________________ The cone viewed from above _____________________________________________________________________ ИЗГЛЕД КЪМ КОНУСА ОТГОРЕ _____________________________________________________________________ Tous ces objets sont des POLYGONES _____________________________________________________________________ It’s OK ; They’re all POLYGONS. _____________________________________________________________________ Всички те са с МНОГО КОНУСИ. _____________________________________________________________________ DANS LE PLAN : La somme des angles d’un - triangle vaut 180° - quadrangle vaut 180° + 180° = 360° - pentangle vaut 180° + 180° + 180° = 540° _____________________________________________________________________ In the PLANE : The angle-sum of a - triangle is 180° - quadrilateral is 180° + 180° = 360° - pentagon is 180° + 180° + 180° = 540° _____________________________________________________________________ На ПОВЪРХНИНАТА: Сумата от ъглита на един: - триъгълник е равна на 180° - четириъгълник е равна на 180° + 180° = 360° - петоъгълник е равна на 180° + 180° + 180° = 540° Etc И т. н.
_____________________________________________________________________ Je craque ... _____________________________________________________________________ I’m goin’ bonkers ... _____________________________________________________________________ Ще се побъркам... _____________________________________________________________________ Et dans le cas du BIANGLE, réduit à un segment, cette somme est nulle _____________________________________________________________________ And for a BIANGLE, which reduces to a line segment, the sum is zero. _____________________________________________________________________ В случая с ДВУЪГЪЛНИКА, сведен до линия, сумата е нула. _____________________________________________________________________ 17 Pourquoi 180° de plus à chaque fois que l’on rajoute un sommet ? _____________________________________________________________________ Why the h’extra 180° every time yer adds a corner ? _____________________________________________________________________ Защо сумата се увеличава със 180° всеки път, когато добавим някой ъгъл? _____________________________________________________________________ Hop !
_____________________________________________________________________ Ha !
_____________________________________________________________________ Хоп!
_____________________________________________________________________ Ceci devrait vous éclairer _____________________________________________________________________ A simple KARATE CHOP will make it clear ... _____________________________________________________________________ Сега трябва да ви просветне... _____________________________________________________________________ Bon, continuons ... _____________________________________________________________________ All right, you two, let’s get on with it. _____________________________________________________________________ Да продължим... _____________________________________________________________________ Je vais maintenant enlever les trois quarts du plan. _____________________________________________________________________ Now I cut out three quaters of the plane. _____________________________________________________________________ Сега ще отрежа три четвърти от плоскостта. _____________________________________________________________________ On dirait une serviette de table _____________________________________________________________________ You might call that a NAPKONE _____________________________________________________________________ Прилича на салфетка. _____________________________________________________________________ Et quand je la regarde par le bout _____________________________________________________________________ And when I look at the end ... _____________________________________________________________________ И когато гледам през нея от единия край... _____________________________________________________________________ Anselme obtient ceci _____________________________________________________________________ Archie sees this _____________________________________________________________________ Ето какво вижда Анселм. _____________________________________________________________________ 18 Sur ce cône, toutes les géodésiques se recoupent elles mêmes (elles se recoupent ici à angle droit). On peut donc y tracer des monoangles _____________________________________________________________________ On such a cone, every geodesic will intersect itself (here at a right angle), so you can draw MONOANGLES.
Върху този конус всички геодезически линии съвпадат (съвпадат под прав ъгъл). Значи можем да начертаем едноъгълници. _____________________________________________________________________ C’était donc vrai ! _____________________________________________________________________ Crikey ! I thought I was jokin’ ! _____________________________________________________________________ Значи съм бил прав. _____________________________________________________________________ Tout dépend de l’angle θ du cône _____________________________________________________________________ It all depends on the angle θ of the cone. _____________________________________________________________________ Всичко зависи от ъгъла θ на конуса. _____________________________________________________________________ Les géodésiques ne se referment pas _____________________________________________________________________ Geodesics don’t close up. _____________________________________________________________________ Геодезическите линии не се затварят. _____________________________________________________________________ cas limite _____________________________________________________________________ Limiting case. _____________________________________________________________________ Краен случай. _____________________________________________________________________ Les géodésiques se referment _____________________________________________________________________ Geodesics close up. _____________________________________________________________________ Геодезическите линии се затварят. _____________________________________________________________________ Каталог: JPP -> telechargeables -> bulgare bulgare -> Joyeuse apocalypse bulgare -> L’aspirisouffle Flight of fancy bulgare -> Всичко е относително jean-Pierre Petit Жан-Пиер Пети bulgare -> Приключенията на анселм лантюрлю énergétiquement vôtre bulgare -> A quoi revent les robots run, robot, run bulgare -> Cosmic Story bulgare -> Les Aventures d’Anselme Lanturlu Изтегляне 0.54 Mb. Сподели с приятели:
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