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