chapter 31

 The Possibility of a “Finite” and yet “Unbounded” Universe

 

But speculations¹ on the structure of the universe also move in quite another direction. The development of non-Euclidean geometry led to the recognition of the fact, that we can cast doubt on the infiniteness of our space without coming into conflict with the laws of thought or with experience (Riemann, Helmholtz). These questions have already been treated in detail and with unsurpassable lucidity² by Helmholtz and Poincaré, whereas I can only touch on them briefly³ here.

 

In the first place, we imagine an existence in two dimensional space. Flat beings with flat implements⁴, and in particular flat rigid⁵ measuring-rods, are free to move in a plane. For them nothing exists outside of this plane: that which they observe to happen to themselves and to their flat “things” is the all-inclusive reality of their plane. In particular, the constructions of plane Euclidean geometry can be carried out by means of the rods e.g. the lattice construction, considered in Section XXIV. In contrast to ours, the universe of these beings is two-dimensional; but, like ours, it extends to infinity⁶. In their universe there is room for an infinite number of identical squares made up of rods, i.e. its volume (surface) is infinite. If these beings say their universe is “plane,” there is sense in the statement, because they mean that they can perform the constructions of plane Euclidean geometry with their rods. In this connection the individual rods always represent the same distance, independently of their position.

 

Let us consider now a second two-dimensional existence, but this time on a spherical⁷ surface instead of on a plane. The flat beings with their measuring-rods and other objects fit exactly on this surface and they are unable to leave it. Their whole universe of observation extends exclusively over the surface of the sphere. Are these beings able to regard the geometry of their universe as being plane geometry and their rods withal as the realisation of “distance”? They cannot do this. For if they attempt to realise a straight line, they will obtain a curve, which we “three-dimensional beings” designate as a great circle, i.e. a self-contained line of definite finite length, which can be measured up by means of a measuring-rod. Similarly, this universe has a finite area that can be compared with the area, of a square constructed with rods. The great charm resulting from this consideration lies in the recognition of the fact that the universe of these beings is finite and yet has no limits.

 

But the spherical-surface beings do not need to go on a world-tour in order to perceive that they are not living in a Euclidean universe. They can convince themselves of this on every part of their “world,” provided they do not use too small a piece of it. Starting from a point, they draw “straight lines” (arcs of circles as judged in three dimensional space) of equal length in all directions. They will call the line joining the free ends of these lines a “circle.” For a plane surface, the ratio of the circumference⁸ of a circle to its diameter, both lengths being measured with the same rod, is, according to Euclidean geometry of the plane, equal to a constant value pi, which is independent of the diameter of the circle. On their spherical surface our flat beings would find for this ratio the value

pi StartStartFraction sine left-parenthesis StartFraction r Over upper R EndFraction right-parenthesis OverOver left-parenthesis StartFraction r Over upper R EndFraction right-parenthesis EndEndFraction comma

i.e. a smaller value than pi, the difference being the more considerable, the greater is the radius⁹ of the circle in comparison with the radius R of the “world-sphere.” By means of this relation the spherical beings can determine the radius of their universe (“world”), even when only a relatively¹⁰ small part of their world-sphere is available for their measurements. But if this part is very small indeed, they will no longer be able to demonstrate that they are on a spherical “world” and not on a Euclidean plane, for a small part of a spherical surface differs only slightly from a piece of a plane of the same size.

 

Thus if the spherical surface beings are living on a planet of which the solar system occupies only a negligibly small part of the spherical universe, they have no means of determining whether they are living in a finite or in an infinite universe, because the “piece of universe” to which they have access is in both cases practically plane, or Euclidean. It follows directly from this discussion, that for our sphere-beings the circumference of a circle first increases with the radius until the “circumference of the universe” is reached, and that it thenceforward gradually decreases to zero for still further increasing values of the radius. During this process the area of the circle continues to increase more and more, until finally it becomes equal to the total area of the whole “world-sphere.”

 

Perhaps the reader will wonder why we have placed our “beings” on a sphere rather than on another closed surface. But this choice has its justification¹¹ in the fact that, of all closed surfaces, the sphere is unique in possessing the property that all points on it are equivalent. I admit that the ratio of the circumference c of a circle to its radius r depends on r, but for a given value of r it is the same for all points of the “world-sphere”; in other words, the “world-sphere” is a “surface of constant curvature.”

 

To this two-dimensional sphere-universe there is a three-dimensional analogy, namely, the three-dimensional spherical space which was discovered by Riemann. its points are likewise all equivalent. It possesses a finite volume, which is determined¹² by its “radius” (2 pi squared upper R cubed). Is it possible to imagine a spherical space? To imagine a space means nothing else than that we imagine an epitome¹³ of our “space” experience, i.e. of experience that we can have in the movement of “rigid” bodies. In this sense we can imagine a spherical space.

 

Suppose we draw lines or stretch strings¹⁴ in all directions from a point, and mark off from each of these the distance r with a measuring-rod. All the free end-points of these lengths lie on a spherical surface. We can specially¹⁵ measure up the area (F) of this surface by means of a square made up of measuring-rods. If the universe is Euclidean, then upper F equals 4 pi r squared; if it is spherical, then F is always less than 4 pi r squared. With increasing values of r, F increases from zero up to a maximum value which is determined by the “world-radius,” but for still further increasing values of r, the area gradually diminishes to zero. At first, the straight lines which radiate from the starting point diverge¹⁶ farther and farther from one another, but later they approach each other, and finally they run together again at a “counter-point” to the starting point. Under such conditions they have traversed the whole spherical space. It is easily seen that the three-dimensional spherical space is quite analogous¹⁷ to the two-dimensional spherical surface. It is finite (i.e. of finite volume), and has no bounds.

 

It may be mentioned that there is yet another kind of curved space: “elliptical space.” It can be regarded as a curved space in which the two “counter-points” are identical (indistinguishable from each other). An elliptical universe can thus be considered to some extent as a curved universe possessing central symmetry.

 

It follows from what has been said, that closed spaces without limits are conceivable. From amongst these, the spherical space (and the elliptical) excels in its simplicity¹⁸, since all points on it are equivalent. As a result of this discussion, a most interesting question arises for astronomers¹⁹ and physicists²⁰, and that is whether the universe in which we live is infinite, or whether it is finite in the manner of the spherical universe. Our experience is far from being sufficient to enable us to answer this question. But the general theory of relativity permits of our answering it with a moderate degree of certainty, and in this connection the difficulty mentioned in Section XXX finds its solution.

 

1 speculations   

n.投机买卖( speculation的名词复数 )；思考；投机活动；推断

参考例句：

• Your speculations were all quite close to the truth. 你的揣测都很接近于事实。 来自《现代英汉综合大词典》
• This possibility gives rise to interesting speculations. 这种可能性引起了有趣的推测。 来自《用法词典》

2 lucidity   

n.明朗，清晰，透明

参考例句：

• His writings were marked by an extraordinary lucidity and elegance of style.他的作品简洁明晰，文风典雅。
• The pain had lessened in the night, but so had his lucidity.夜里他的痛苦是减轻了，但人也不那么清醒了。

3 briefly   

adv.简单地，简短地

参考例句：

• I want to touch briefly on another aspect of the problem.我想简单地谈一下这个问题的另一方面。
• He was kidnapped and briefly detained by a terrorist group.他被一个恐怖组织绑架并短暂拘禁。

4 implements   

n.工具( implement的名词复数 )；家具；手段；[法律]履行（契约等）v.实现( implement的第三人称单数 )；执行；贯彻；使生效

参考例句：

• Primitive man hunted wild animals with crude stone implements. 原始社会的人用粗糙的石器猎取野兽。 来自《现代汉英综合大词典》
• They ordered quantities of farm implements. 他们订购了大量农具。 来自《现代汉英综合大词典》

5 rigid   

adj.严格的，死板的；刚硬的，僵硬的

参考例句：

• She became as rigid as adamant.她变得如顽石般的固执。
• The examination was so rigid that nearly all aspirants were ruled out.考试很严，几乎所有的考生都被淘汰了。

6 infinity   

n.无限，无穷，大量

参考例句：

• It is impossible to count up to infinity.不可能数到无穷大。
• Theoretically,a line can extend into infinity.从理论上来说直线可以无限地延伸。

7 spherical   

adj.球形的；球面的

参考例句：

• The Earth is a nearly spherical planet.地球是一个近似球体的行星。
• Many engineers shy away from spherical projection methods.许多工程师对球面投影法有畏难情绪。

8 circumference   

n.圆周，周长，圆周线

参考例句：

• It's a mile round the circumference of the field.运动场周长一英里。
• The diameter and the circumference of a circle correlate.圆的直径与圆周有相互关系。

9 radius   

n.半径，半径范围；有效航程，范围，界限

参考例句：

• He has visited every shop within a radius of two miles.周围两英里以内的店铺他都去过。
• We are measuring the radius of the circle.我们正在测量圆的半径。

10 relatively   

adv.比较...地，相对地

参考例句：

• The rabbit is a relatively recent introduction in Australia.兔子是相对较新引入澳大利亚的物种。
• The operation was relatively painless.手术相对来说不痛。

11 justification   

n.正当的理由；辩解的理由

参考例句：

• There's no justification for dividing the company into smaller units. 没有理由把公司划分成小单位。
• In the young there is a justification for this feeling. 在年轻人中有这种感觉是有理由的。

12 determined   

adj.坚定的；有决心的

参考例句：

• I have determined on going to Tibet after graduation.我已决定毕业后去西藏。
• He determined to view the rooms behind the office.他决定查看一下办公室后面的房间。

13 epitome   

n.典型，梗概

参考例句：

• He is the epitome of goodness.他是善良的典范。
• This handbook is a neat epitome of everyday hygiene.这本手册概括了日常卫生的要点。

14 strings   

n.弦

参考例句：

• He sat on the bed,idly plucking the strings of his guitar.他坐在床上，随意地拨着吉他的弦。
• She swept her fingers over the strings of the harp.她用手指划过竖琴的琴弦。

15 specially   

adv.特定地；特殊地；明确地

参考例句：

• They are specially packaged so that they stack easily.它们经过特别包装以便于堆放。
• The machine was designed specially for demolishing old buildings.这种机器是专为拆毁旧楼房而设计的。

16 diverge   

v.分叉，分歧，离题，使...岔开，使转向

参考例句：

• This is where our opinions diverge from each other.这就是我们意见产生分歧之处。
• Don't diverge in your speech.发言不要离题。

17 analogous   

adj.相似的；类似的

参考例句：

• The two situations are roughly analogous.两种情況大致相似。
• The company is in a position closely analogous to that of its main rival.该公司与主要竞争对手的处境极为相似。

18 simplicity   

n.简单，简易；朴素；直率，单纯

参考例句：

• She dressed with elegant simplicity.她穿着朴素高雅。
• The beauty of this plan is its simplicity.简明扼要是这个计划的一大特点。

19 astronomers   

n.天文学者，天文学家( astronomer的名词复数 )

参考例句：

• Astronomers can accurately foretell the date,time,and length of future eclipses. 天文学家能精确地预告未来日食月食的日期、时刻和时长。 来自《简明英汉词典》
• Astronomers used to ask why only Saturn has rings. 天文学家们过去一直感到奇怪，为什么只有土星有光环。 来自《简明英汉词典》

20 physicists   

物理学家( physicist的名词复数 )

参考例句：

• For many particle physicists, however, it was a year of frustration. 对于许多粒子物理学家来说，这是受挫折的一年。 来自英汉非文学 - 科技
• Physicists seek rules or patterns to provide a framework. 物理学家寻求用法则或图式来构成一个框架。