In the Pythagorean theorem episode of NBC Learn's "The Science of NFL Football," you see that a defender in the middle of the field must take the proper angle of pursuit to catch a ball-carrier making a dash down the sideline for the end zone.
In chasing down the ball-carrier, the defender basically runs along the diagonal of a right triangle, in which the sum of the squares of the sides equals the square of that diagonal. You might know this relation, discovered by the 5th-century B.C. Greek mathematician Pythagoras, as a² + b² = c².
The "c " is the hypotenuse, and although it represents the longest side of a right triangle, it is the shortest path between the two points on either end. If the points on the triangle were places to visit in a city, you probably wouldn't bother walking along a and b if you could directly take c.
But the hypotenuse isn't always the shortest route. In fact, it is only the shortest one on football fields and other flat surfaces. On spheres and other shapes, it may not be.
You can see this distinction if you draw a right triangle on a globe. First, let's pick a city on the equator--for simplicity, say it's Quito, Ecuador, on the Pacific coast of South America. From Quito, trace a longitude line to the north pole; then make a 90-degree turn to the right and head straight back down. At the equator, you'll notice a city nearby called Libreville, the capital of the country of Gabon in Africa.
Now draw a line along the globe's surface starting at Quito and going toward Libreville. You probably went eastward, passing over Brazil and the Atlantic Ocean. Indeed, this hypotenuse, traversing one quarter of the globe, marks the shortest distance. But that is not the only hypotenuse.
Mathematically speaking, you would still have a right triangle if you went westward from Quito, circumnavigating Earth along the equator to get to Libreville. The hypotenuse in this case is three fourths of the circumference. It would have been shorter to travel from Quito to the North Pole and then down to Libreville.
The Pythagorean theorem only works on two-dimensional surfaces like football fields; mathematicians refer to such surfaces as Euclidean geometry (named for Euclid, the 3rd-century B.C. Greek mathematician). The theorem fails for non-Euclidean geometries, such as spheres and more complex geometries like saddles. Indeed, all the rules you learned in school, like parallel lines staying parallel, only refer to Euclidean geometry. In the non-Euclidean universe, parallel lines may actually diverge or converge.
Although non-Euclidean geometry may seem exotic and unfamiliar, it is actually common in many fields of science--perhaps most notably, in Einstein's general relativity theory, in which gravity can bend the shape of space and time.
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28 Comments
Add CommentJolly good. Here's another conundrum on akin to zeno's. If I indent the hypotenuse of the 3-4-5 triangle with a set of small steps each parallel and in proportion to the 3-4 sides, the sum of their horizontal and vertical parts is still 7 units. As I make the steps smaller their sum is still 7. From a long distance they merge into the hypotenuse. At which point does the sum of the digital/quantum steps become 5?
Reply | Report Abuse | Link to thisso whats the answer?
Reply | Report Abuse | Link to thisIs this true of triangles on spheres. What relationship holds there - and on saddles. Does it imply something for space time?
Could something like this happen where the magnetic and electrical field of a particle were normal and the direction of travel of the particle was travelling within a dimension which didn't take the fields with them proportionally?
Reply | Report Abuse | Link to thisFrom a long distance they merge into the hypotenuse. That is a visual effect. The actual measurements don't change...do they...or did I miss your point?
Reply | Report Abuse | Link to thisMany predators (eg. wolves, bears catching salmon, your dog catching a frisbie) inherit or learn the behavior of mental estimation which results in successful pursuit. Perhaps someone has (or will) discover the "Pythagorean gene" that makes this mental math so intuitive?
Reply | Report Abuse | Link to thisYou can't draw a triangle on the surface of a sphere. A triangle is composed of straight segments. Connecting three points along the surface of a sphere gives you 3 dots connected by curved segments. Even if there is a right angle at one of the points, there is no hypotenuse, as there is no triangle (so let's call it a "pseudo-hypotenuse"). To connect 3 points with straight lines to form a triangle, on the surface of a sphere, you need the lines to go through the sphere, not stay on the surface. You will find that the shortest distance between the points is... a straight line.
Reply | Report Abuse | Link to thisThe point about the 'pseudo-hypotenuse' on the surface of a sphere not necessarily being the shortest route, is irrelevant to the title of the article, because it is NOT a straight line. It’s a curved line.
You’ve come up with a complicated way of saying “a straight line isn’t always the shortest distance between two points, if the line in question isn’t actually straight.”
This is only true in "R3", a three dimensional space. A curved 2d space (R2) OTOH is what the author was discussing. In this case the SPACE is curved, but the lines are straight, a straight line being defined as the shortest distance between two points in that space. R3 and higher dimensional spaces can also have curvature, which is analogous to those seen in R2. A positively curved space is like a sphere, finite in extent but unbounded. Negatively curved spaces are often said to be saddle shaped in R2 and are either infinite or bounded.
Reply | Report Abuse | Link to thisYou can easily reconstruct geometry in these kinds of spaces too and propose axioms analogous to those of the familiar Euclidean (flat) geometry. These geometries are just as internally consistent and valid as Euclidean geometry. For instance geometry on the surface of the Earth confirms that in a positively curved space triangles angles sum to less than 180 degrees.
The shortest distance between two points is a geodesic. In two-dimensional space, a geodesic is a straight line. Euclidean geometry is geometry in two-dimensional space. It is surprising that non-euclidean geometry was developed only in the 19th century by Gauss and Riemann bec. Ptolemy in the 2nd century already knew spherical trigonometry which is actually 3-dimensional (non-euclidean) geometry.
Reply | Report Abuse | Link to thisWhat is straight line anyway? One can easily make a triangle on a (FLAT?) surface with curved lines whose sum is 180 degrees. And what is flat? One can also have a coordinate system in which a single dimension is is represented by a curved line. On surface of earth if one digs a tunnel straight line will still be the shortest distance. In space straight line is still the shortest distance but general relativity does not allow us to travel that path and no one has as yet proved the general relativity to break. We therefore accept the notion that light travels the shortest distance in free space. That is how we define the shortest distance and give physics precedence over mathematics(Euclid). Curvature of space time is a concept and then one should also consider space and time are relative and differ from observer to observer i.e. every you and me has his own space time.
Reply | Report Abuse | Link to thisIn Euclidean geometry, a line is always straight. If it is not straight, it is not a line. It is a curve. This definition is consistent with analytic geometry - a line is represented by a linear equation. It is always straight. A curve is represented by non-linear equations such as quadratic, cubic, quartic and polynomials. Well, a curve is not a line. A straight line is redundant like a spherical ball.
Reply | Report Abuse | Link to thisThere is no straight line in space bec. space itself is curved. Even light follows the curvature of space. Light travels the shortest distance, a 3-dimensional geodesic, but it's not a (straight) line.
@ tharter, who said: "...a straight line being defined as the shortest distance between two points in that space."
Reply | Report Abuse | Link to thisInteresting, but with that definition, the title of the article is still irrelevent.
And I suppose the surface of a football field is curved because it follows the surface of the spherical Earth? More likely it was leveled using the shortest distance between two points.
Reply | Report Abuse | Link to thisFortunately, a football player can ignore all the superfluous constraints and, unlike the illustration, begin his pursuit of the receiver several yards away from the line of scrimmage or following an initial collision. Representative generalizations seldom occur in reality.
Btw, perhaps this inadequate academic idealization of real world problem analysis is the reason why so many pursuers end up tackling the spacetime behind Chris Johnson...
Reply | Report Abuse | Link to thisA line is the shortest distance between 2 points in a flat manifold of any n dimensional space. In a curved manifold, the shortest distance is of course a curve. In curved manifolds, a triangle does not have 180 deg. Flat means zero curvature.
Reply | Report Abuse | Link to thisIn principle, we cannot draw a straigth line or a triangle with 180 deg. bec. our universe is a curved manifold. But the universe is so big compared to earth that all our measurements cannot detect the deviation.
We can do Euclidean geometry on a piece of paper bec. the paper is so small compared to the earth's surface. The error is undetectable. But if your paper is as large as North America, you can detect the error. Your triangle does not give 180 deg. It's non-euclidean geometry.
Same is true with earth and the universe. The latter has a radius of curvature >70 billion light-yrs. Most astronomers think the universe is a 3-sphere (4-dimensional sphere). The famous expanding balloon model is a 3-sphere. The volume of a 3-sphere with 70 gly radius is 6,770,580 billion cubic light-yrs. This is well beyond the visible universe. So the universe is not infinite but it's REALLY BIG.
In curved space time angles of triangle may not always be 180 degrees but it is possible to have triangle with 180 degrees and that will also follow cosine formula. Now the question is how can one define straight or non-straight(curved) in presence of force like gravity. Further what is a dimension anyway? One can represent a curved dimension with a single variable. Forget Euclidean and Gaussian what is their in name, if you call an owl a cat it makes no difference as far as their existence, their physical property and every thing else is concerned.
Reply | Report Abuse | Link to thisIn spherical trigonometry, triangles follow the spherical law of cosines but they never equal 180 deg.
Reply | Report Abuse | Link to thisGlobally, spacetime is curved. This curvature may be too small to be detectable locally. Spacetime may look perfectly flat locally. The presence of mass distorts spacetime. This is what we call gravitational field. The distortion is a pronounced curvature of local spacetime.
Dimensions are length, width and height in 3-dimensional geometry. A curve, just like a line, has length. But a curve is not a line. They have different properties. A curve has variable slope. A line has constant slope. They are represented by different types of equation.
We have to study Euclid and Gauss bec. geometry has already been invented. It's easy to get lost trying to reinvent it.
Suppose I am near a black hole and I shoot photons or light beam parallel to the horizon. Now the beam will return to me after fraction of second. Now without moving an inch does that mean I am travelling in time. Similarly if I could reach a place faster than light with same initial position in space time for both me and light does that mean I am travelling in time.
Reply | Report Abuse | Link to thisAnyway I will have to use a force to balance the effect of gravity and I suppose my spaceship uses thermonuclear energy to do that.
Reply | Report Abuse | Link to thisRegarding a slope of geodesic path in isolated space on which I am moving I can never determine that slope I will always move straight.
Reply | Report Abuse | Link to thisAnd Dr.Strangelove why are you obsessed with spheres. Geodesic surfaces can be curved, zig-zag, concave-convex with each other in space. Had Mr. Einstein been alive I would have surely discussed this with him.
Reply | Report Abuse | Link to thisOh one more thing I was talking of cosine formula with Pythagorean theorem as special case and not direction cosines.
Reply | Report Abuse | Link to thisIf you travel in space, you also travel in time bec. spacetime is inseparable. That's special relativity.
Reply | Report Abuse | Link to thisYou can calculate the slope of geodesics without leaving the manifold if you know the radius of curvature and shape of the manifold. That's what cosmologists do. They make calculations about the size and shape of the universe without leaving the universe.
A generalized Pythagorean theorem is applicable to flat manifolds. It defines the length of a line in flat manifolds of any n dimensional space.
I used spherical triangles bec. they are the curved manifold easiest to visualize. Higher dimensional curved manifolds are impossible to visualize but their properties are computable.
I'm sure Mr. Einstein would love to talk to you.
Only in slow moving things are there 2 points.
Reply | Report Abuse | Link to thisAs soon as the things move faster, 2 thing are
1 thing and visa versa.
Remember that there is a major fundamental difference between SPACE (distance) and TIME.
When 2 points cool to theoretical absolute zero,
this does no mean there is no TIME.
Be sure what quantity (mixed) you are measuring TIME or SPACE (distance) before there is any type of math and geometrical area between the 2 so called points.
Conjecture, at ZERO TIME there is no SPACE between
2 points.
Look at this simple problem. Quantity 1 TIME and Quantity 2 SPACE or distance.
Reply | Report Abuse | Link to thisAt absolute zero degrees Kelvin, there is still TIME.
What space is between 2 points at absolute zero degree Kelvin is there?
The space between 2 points is obviously relative to the heat or energy between 2 points.
Classical mechanics turns thru themodynamics into quantum mechanics where at where at 1 point there
is only probablasitic SPACE TIME between 2 points and
elswhere, there is no SPACE (ecludian distance between 2 points) yet TIME is still existence and has not changed, to longer or shorter.
They teach us this is Physics 101. What is the mystery?
I skipped Physics 101, so you'd have to explain to me in more detail how space disappears at absolute zero. I cannot accept that assertion at face value. In a continuously expanding universe, its temperature would eventually reach absolute zero. So at that moment space would disappear?
Reply | Report Abuse | Link to thisThe article is misleading and inaccurate. In non-euclidean geometry, a straight line is defined as a geodesic, the shortest distance bet. 2 points in any n dimensional space. So by definition, a straight line is necessarily the shortest distance. Otherwise, it is a curve.
Reply | Report Abuse | Link to thisIn the example given, the hypotenuse of the right triangle is not the shortest distance bec. it is a curve, not a line. We have to define what are lines and curves in spherical trigonometry, which is non-euclidean geometry.
On earth, longitudes are geodesics (lines) and latitudes are non-geodesics (curves). This may not be obvious but if you pull a string bet. 2 points on earth's surface to get the shortest distance, the string will trace a longitude and deviate from a latitude. Btw, longitudes are non-parallel lines that pass through the poles wherever they are located on the sphere.
To see why geodesics are lines and non-geodesics are curves, we have to use projective geometry. If you do perpendicular projection longitudes on a plane surface, longitudes trace a straight line on a plane surface and latitudes trace a curve.
In short, if you draw a right triangle on a sphere, its sides are not the shortest distances bet. the 3 points bec. the sides are not lines but curves. They may look straight but they are not. To get a triangle whose sides are lines, you have to draw a spherical triangle whose angles are >180 deg., which is not a right triangle whose angles are =180 deg.
The fundamental issue here is not whether a line is the shortest distance bet. 2 points but what appears to be a line in curved surfaces may actually be a curve that only looks straight. The shortest distance is a geodesic that corresponds to a straight line when projected to a plane geometry.
The shortest distance between two points is 0.
Reply | Report Abuse | Link to thisWhy put anything like a line or curve in the way or 2 points?
Your not debating the "Shortest" distance , you're debating the most direct route.
On second thought the most direct route would also be 0. But you would have to bend the plane between to merge the 2 points, to remove the distance between the 2 points. ;-)
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