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(M-6a) Beyond Pythagoras
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The proof to the theorem of Pythagoras in the classical Greek geometry of Euclid is quite different from the way it is done in the preceding section. Euclid's proof, still taught in many classrooms, is purely geometrical. It constructs a square on the outside of each side of the right-angle triangle, and then, using the definition of area and theorems based on it, goes to show that the sum of the areas of the squares above the shorter sides equals the area of the third, bigger square.
It is the same theorem, but with one significant difference: nowhere is the length of any line assigned a numerical value. It is pure geometry with no algebra. Likewise, other parts of Euclid's geometry deal with lines, angles and areas, but never assign to them values in numbers. Of course, Greek mathematicians also knew about applying numbers to geometry. Builders modeled wall corners by building triangles out of three wood pieces with lengths (3, 4, 5) times some convenient reference length (or ruler), which produce an exact right angle (that example is given at the beginning of the previous section). Expressing length in numbers allowed the application of the powerful rules of algebra, but the creators of Euclid's geometry tried to do without numbers, feeling perhaps that a geometry without numbers was more refined, more conceptual. The study of geometry with numerical distances (also angles and areas) was instead relegated to trigonometry, literally "the measurement of triangles." Greek mathematics developed its own brand of trigonometry, credited to Hipparchus of Nicea (Nikeh-ya), who measured extensively the positions of stars on the night sky, discovered the precession of the equinoxes, and used a solar eclipse to estimate the distance to the Moon. "Greek" here should mean the entire Greek-speaking culture of the eastern Mediterranean, extending far beyond Greece: Euclid and Eratosthenes, for instance, lived in Alexandria of Egypt. Coordinate Geometry![[Image:(x,y)]](Sfigs/Sxy.gif)
All these proofs are correct, but practical applications are all based on numbers. When Descartes formulated coordinate geometry (or "cartesian geometry" after the discoverer's name, also "analytic geometry"), he based it on the Pythagoras theorem expressed in numbers. He labeled each point P on a plane by its perpendicular distances (x,y) from two perpendicular axes (its "cartesian coordinates"), as described on the web page linked here. The "origin" of the coordinates is the point (0, 0), and by the theorem of Pythagoras, the distance r of the given point from the origin satisfies
A point in 3-dimensional space can be labeled by its perpendicular distances (x,y,z) from three mutually perpendicular axes. One can apply the theorem twice, first to (x,y) giving the distance ρ (Greek rho) of the projection of P pn the (x,y) plane, then again to (ρ,z) in the plane defined by P and the z-axis: ρ2 = x2 + y2 r2 = ρ2 + z2 This makes r the distance from the origin point (0, 0, 0) , and it follows from the above applications of the Pythagoras theorem that ![[IMAGE:3-D Coord]](Sfigs/S3D.gif) r2 = x2 + y2 + z2 (2) If one is limited to a 2-dimensional curved surface, like the surface of a sphere, each point can be labeled by its perpendicular distances X (on the surface!) from the equator (measured along a line of longitude), and Y from some reference line of longitude (measured similarly at a constant latitude). Because of the curvature, it no longer holds true that the distance R from the intersection satisfies equation (1), instead one observes thatalthough the relation comes close to equality if X and Y are very small. Indeed, one can define the "flatness" of a surface by requiring the Pythagoras condition (1) to hold between for any point and any origin, and a space is said to be flat, if condition (2) holds there in a similar way. Because of flatness, if we want the distance between any other two other points, we simply go over to new coordinates, in which the coordinates (x1,y1,z1) of one of the points are subtracted from any "old" ones, giving a new cartesian system with its origin at the selected point. (In curved coordinates, transforming to a system with a new origin is more complicated.) This is a completely new way of looking at the theorem of Pythagoras--not as a mathematical result but as a condition required of surfaces or spaces to be "flat." Is the space inhabited by us flat? Experimentally--in the lab, and even at greater observable distances--yes. But as in measurements on the surface of a sphere, flatness may be close to perfect on the small scale but not at great distances, where the prescription for distance ("the metric tensor") may be distorted. Einstein's general relativity in fact proposes that gravity can deform the metric. Is space 4-dimensional?The question here is not whether we live in a 3-D space embedded in a 4-D one. Such embedding means that the 3-D metric we observe may contain all sorts of curvature, just as a curved 2D surface (for instance, the surface of a sphere) may be embedded in a larger 3-D flat space. Two-dimensional creatures living on that surface would be unaware of the 3D space around them, they would be unable to observe 3-D objects, but just observe that in their universe the theorem of Pythagoras only holds for points very close to each other, not for large separations. Our observable universe is curved in such a way, which allows it to continue the expansion started at the "big bang."No, the question here is--is time a 4th dimension? Yes and no. It is complicated, and requires re-examining what is meant by "number." The concept of numbers starts with integers 1,2,3... which can be added and multiplied, later expanded to also include negative numbers (-1,-2,-3...) and zero, with relations like (3) + (-3) = 0 Next simple fractions like (1/3) may be added, (3) . (1/3) = 1 and then "rational numbers" like 5/7, 65/7, 0.12345, -1.55 Still later numbers which cannot be expressed as fractions may also be included, like the square root of 2, most trigonometrical functions and most logarithms (these are dealt with further along in this "math refresher"). Objects that satisfy this widest definition are known as real numbers. Real numbers are all one needs for labeling any distance along a line. However, mathematicians are an inventive lot and have devised all sorts of other mathematical objects obeying different kinds of rules (for instance, with the product of two "numbers" depending on the order in which it is performed) . Such systems are generally labeled "algebras"--vector algebra, matrix algebra, etc. They have arisen from all sorts of applications, such as handling the "attitude" of a satellite in space (= the direction in which it points). One property of real numbers is that multiplying a number by itself always gives a positive result: No real number has the square (-1)! One of the oldest "expanded number systems" adds an imaginary number denoted as i, with the property From this a wider range of "complex numbers" (a + bi) can be defined, where a and b are any real numbers. Some electrical engineers prefer "j" because the letter i may be used to denote electrical current. It turns out that practically any calculation with real numbers has its equivalence with complex numbers, with interesting new twists (e.g defining 22i+3). The only notable difference is that among two positive numbers, one is always larger, and if one adds the smaller one to itself again and again, the result is ultimately larger than the other or any positive number one can name. There is no analogous property among complex numbers. This means assigning an imaginary length to a stick has no meaning: how does one compare a stick of length (4i) to a stick of length 4 ? However, examining the laws of mechanics and electricity brings out an interesting symmetry: in many of them time behaves as if it were a 4th dimension, but with imaginary values. Well... There exists one problem here: time t is measured in seconds, while distance x is measured in meters, or other units of length. How can one compare seconds to centimeters? The laws of physics indeed suggest that the 4th dimension is ct, time multiplied by the velocity of light c, about 300,000 kilometers per second: multiplying kilometers-per-second by time-in-seconds gives a distance. Of course, time being "imaginary", one still has to multiply by i = √−1 , which makes the 4th dimension ict and provides (for instance) a natural formulation for wave phenomena. In this sort of 4-dimensional "spacetime" the position of a grain of sand observed at (x,y,z) at time t is (x,y,z,ict) and its "distance" from the "origin" (0,0,0,0) of spacetime is formally satisfied by the equation of Pythagoras s2 = x2 + y2 + z2 − c2t2 (5) Don't try too hard to interpret this! The message at this level is simply that while time is a 4th dimension, it needs to be measured in imaginary numbers, making it a "time-like" dimension. And the difference between distances and time periods is like the difference between ordinary numbers and imaginary ones.Or apples and oranges, if you will. Note: Actually, time has an additional complication. In the basic laws of physics, going forwards in time looks very similar to going backwards in time. The reason one cannot actually go back in time is the imposition in nature of the second law of thermodynamics, by which observed time always flows in the direction in which a quantity known as entropy increases. This is a complicated area, not followed here. |
Next Stop: #M-7 Trigonometry: What is it good for?
Author and Curator: Dr. David P. Stern
Mail to Dr.Stern: stargaze("at" symbol)phy6.org .
Updated 22 April 2013, again 27 October 2016.
