Author: George Gamow
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Is there any sense in asking: “Is the number of all numbers larger or smaller than the number of all points on a line?”
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Which do you think is larger: the number of all numbers, both even and odd, or the number of even numbers only?
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In fact in the world of infinity a part may be equal to the whole!
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Following Cantor’s rule for comparing two infinities, we can also prove now that the number of all ordinary arithmetical fractions like or is the same as the number of all integers.
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“Well, it is all very nice,” you may say, “but doesn’t it mean simply that all infinities are equal to one another? And if that’s the case, what’s the use of comparing them anyway?” No, that is not the case, and one can easily find the infinity that is larger than the infinity of all integers or all arithmetical fractions.
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there is the same number of points in lines one inch, one foot, or one mile long.
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two infinities of points are equal. A still more striking result of the analysis of infinity consists in the statement that: the number of all points on a plane is equal to the number of all points on a line.
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A still more striking result of the analysis of infinity consists in the statement that: the number of all points on a plane is equal to the number of all points on a line.
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the number of all points on a plane is equal to the number of all points on a line.
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In a similar way it is easy to prove also that the infinity of all points within a cube is the same as the infinity of points within a square or on a line.
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variety of all possible curves, including those of most unusual shapes, has a larger membership than the collection of all geometrical points, and thus has to be described by the third number of the infinite sequence.
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infinite numbers are denoted by the Hebrew letter ℵ (aleph)
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infinite numbers are denoted by the Hebrew letter ℵ (aleph) with a little number in the lower right corner that indicates the order of the infinity.
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One large system of mathematics, however, has up to now managed to remain quite useless for any purpose except that of stimulating mental gymnastics, and thus can carry with honor the “crown of purity.” This is the so-called “theory of numbers”
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Thus the problem of finding a general formula by the application of which only primes may be produced is still unsolved.
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Another interesting example of a theorem of the theory of numbers that has been neither proved nor disproved is the so-called Goldbach conjecture, proposed in 1742, which states that each even number can be represented as the sum of two primes.
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This table shows first of all that the relative number of primes decreases gradually as the number of all integers increases, but that
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This table shows first of all that the relative number of primes decreases gradually as the number of all integers increases, but that there is no point at which there are no primes.
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the relative number of primes decreases gradually as the number of all integers increases, but that there is no point at which there are no primes.
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the percentage of primes within an interval from 1 to any larger number N, is approximately stated by the natural logarithm of N.
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mathematicians are obstinate people, and when something that seems to make no sense keeps popping up in their formulas, they will do their best to put sense into it.
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When we multiply a real number, say 3, representing a point on the horizontal axis, by the imaginary unit i we obtain the purely imaginary number 3i, which must be plotted on the vertical axis. Hence, the multiplication by i is geometrically equivalent to a counterclockwise rotation by a right angle.
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Thus the statement that the “square of i is equal to −1” is a much more understandable statement than ‘turning twice by a right angle (both turns counterclockwise) you will face in the opposite direction.”
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Another hidden treasure that was found by using the imaginary square root of −1 was the astonishing discovery that our ordinary three-dimensional space and time can be united into one four-dimensional picture governed by the rules of four-dimensional geometry.
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the fact is that a great many of the most fundamental properties of space do not require any measurements of lengths or angles whatsoever. The branch of geometry concerned with these matters is known as analysis situs or topology19 and is one of the most provocative
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the fact is that a great many of the most fundamental properties of space do not require any measurements of lengths or angles whatsoever. The branch of geometry concerned with these matters
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the fact is that a great many of the most fundamental properties of space do not require any measurements of lengths or angles whatsoever. The branch of geometry concerned with these matters is known as analysis situs or topology
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Apparently then, V+F=E+2 is a general mathematical theorem of a topological nature since the relationship expression does not depend on measuring the lengths of the ribs, or the areas of the faces, but is concerned only with the number of the different geometrical units (that is, vertices, edges, faces) involved.
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One interesting consequence of Euler’s formula is the proof that there can be only five regular polyhedrons, namely those shown in Figure 14.
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We have limited ourselves only to the polyhedrons that, so to speak, do not have any holes through them;
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For the doughnut-shaped, or, speaking more scientifically, torus-shaped, polyhedrons we have V+F=E, whereas for the “pretzel” we have V+F=E−2. In general V+F=E+2−2N where N is the number of holes.
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it has been conclusively proved that seven different colors are enough to color any possible combination of subdivisions of a doughnut without ever coloring two adjacent sections the same, and examples have been given in which the seven colors are actually necessary.
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Thus the three-dimensional generalization of the map-coloring problem can be formulated somewhat as follows: We are asked to build a space mosaic using many variously shaped pieces of different materials, and want to do it in such a way that no two pieces made of the same material will be in contact along the common surface.
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Thus the three-dimensional generalization of the map-coloring problem can be formulated somewhat as follows: We are asked to build a space mosaic using many variously shaped pieces of different materials, and want to do it in such a way that no two pieces made of the same material will be in contact along the common surface. How many different materials are necessary?
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Let us first try to build a model of a three-dimensional space that would have properties similar to the surface of a sphere. The main property of a spherical surface is, of course, that, though it has no boundaries, it still has a finite area; it just turns around and closes on itself. Can we imagine a three-dimensional space that would close on itself in a similar way, and thus have a finite volume without having any sharp boundaries?
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One must rather think about an apple with an intricate system of channels eaten through it by worms. There must be two breeds of worm, say white and black ones, who do not like each other and never join their respective channels inside the apple although they may start them at adjacent points on the surface.
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We are not yet quite through with the apple and the worms, and the next question we ask is whether it is possible to turn a worm-eaten apple into a doughnut.
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Your body also has the shape of a doughnut, though you probably never thought about it. In fact, in the very early stage of its development (embryonic stage) every living organism passes the stage known as “gastrula,” in which it possesses a spherical shape with a broad channel going across it.
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If you compare two gloves of a pair (Figure 21) you will find them identical in all measurements and yet there is a great difference since you cannot put the left glove on the right hand or vice versa. You can turn and twist them as much as you like, but still the right glove remains right, and the left glove remains left.
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If the object does not possess a plane of symmetry, and is as we say, asymmetrical, it will be bound in two different modifications—a right- and a left-handed one.
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By way of analogy one could say that a right glove can be turned into a left glove by taking it out of our space in the fourth direction and rotating it in a proper way before putting it back.
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This surface has many peculiar properties, one of which can be easily discovered by cutting, with a pair of scissors, completely around it in a line parallel to the edges (along the arrows in Figure 23). You would expect, of course, that by doing so, you would cut the ring into two separate rings. Do it, and you will see that your guess was wrong: instead of two rings you will find only one ring, but one twice as long as the original and half as wide!
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In short, by walking around the surface of Möbius, our “left-profile” donkey has turned into one with a “right profile.”
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The Möbius strip shown in Figure 23 represents a part of a more general surface, known as the Klein bottle