Hands-on Geometry Explorations

St. Patrick’s College, Drumcondra

Hands-on Geometry Explorations Colm Mulcahy Spelman College

10 March 2009

Philosophy Tell me and I’ll forget. Show me and I may not remember. Involve me and I’ll understand. (Native American saying)

Geometry is the user interface of mathematics (John Horton Conway)

c 2 = a2 + b 2 ? Everybody is familiar with Pythagoras’ theorem relating the sides of a right angle triangle: c 2 = a2 + b 2 . Who has not been asked to use this to work out the hypotenuse of a triangular field whose other sides measure 300 metres by 400 metres? Yet the tradional answer of 500 metres is incorrect! The reason is simple: We live on the surface of a sphere, not on a plane. Ironically, people who navigate on board (air)planes need to be able to answer such questions precisely. Few of the rest of us have the training required to deal with these issues.

Straight Lines on a Sphere

I could be bounded in a nutshell and count myself a king of infinite space. (Hamlet) Spherical geometry vanished from the curriculum at all levels many decades ago, and is only beginning to make a comeback. The planar Pythagoras’ theorem is a special case of: c 2 = a2 + b 2 − 2ab cos(γ). We’ll show how to derive the spherical version from a corresponding Cosine Rule for spherical triangles.

Straight lines on a sphere

What planar geometry works on spheres? Before we can talk about triangles, we need to agree just what constitutes a side! What is a straight line on a sphere? The shortest distance between two points? How does one measure/check that? Can physics help?

Straight lines on a sphere A straight line (=geodesic) on a sphere is an arc of a great circle. Great circles have minimal curvature (and maximal radius) among all circles on the surface of a sphere. ◮

Two points determine more than one geodesic segment.

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Distinct geodesics meet twice. The shaded area is a lune.

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There are no parallels. (Given a line L and a point not on L, no lines exist that contain the point, and do not meet L.)

Spherical Triangles Given 3 points, do we get a triangle by connecting to get 3 sides? There are infinitely many ways to connect 2 antipodal points. There are 2 ways to connect 2 distinct points on a sphere in general, so given 3 “non-collinear” points, no 2 of which are antipodal, there are 23 = 8 ways to form a triangle from them! How do we decide which one is the most reasonable? Or which zone is the inside and which is the outside?

Small spherical triangles Maybe only consider “small” triangles. Meaning . . . ?

These are determined by six angles (and the sphere’s radius): three vertex angles α, β, γ, and three side angles a, b, c.

The side angles here replace the side lengths of the planar case. Of course a =

|a| r ,b

=

|b| r ,c

=

|c| r ,

in radians.

Small spherical triangles Small spherical triangles can be cut from paper or cardboard discs, like pizza pieces. Key observation: for fixed side lengths |a|, |b|, |c|, we have a, b, c, → 0 as r → ∞, yielding a planar triangle. In this case sin(a) → a and cos(a) → 1 etc. Two questions worth asking about any spherical triangle fact: ◮

Is it understandable/believable as the radius goes to infinity?

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Is it understandable/believable for right angle triangles?

Planar Pythagoras’ Theorem

Pythagoras’ theorem relates the sides of a right angle planar triangle: c 2 = a2 + b 2 where c is the hypotenuse. There are many revealing ways ways to view/prove this, and it is of course a special case of the Cosine Rule: c 2 = a2 + b 2 − 2ab cos(γ) Is there a Cosine Rule for (small) spherical triangles? Yes!

The Cosine Rule on a sphere

The Cosine Rule for Sides: cos(c) = cos(a) cos(b) + sin(a) sin(b) cos(γ) yielding the Pythagorean cos(c) = cos(a) cos(b) when γ = 90o ! In terms of side lengths, this says: cos(

|c| |a| |b ) = cos( ) cos( |) r r r

c 2 = a2 + b 2? revisited

Now we can solve a problem like: A right-angled triangular plot of land measures 300 and 400 kilometres respectively on its short sides. How long is the hypothenuse? Is it longer or shorter than 500 kilometres? (Assume r = 6370 kilometres.) Answer: 499.88 kilometres!

Small right angled spherical triangles Assume vertex angle γ = 90o . We can derive other several key identities, among them: cos(α) = tan(b) cot(c) cos(α) = cos(a) sin(β) sin(b) = tan(a) cot(α) cos(c) = cot(α) cot(β) Also: sin(α) = cos(α) =

sin(a) and sin(c) sin(b) cos(a), sin(c)

Alternative Pythagoras yielding the apparently asymmetrical sin2 (c) = sin2 (a) + sin2 (b) cos2 (a). Since we clearly also have: sin2 (c) = sin2 (a) cos2 (b) + sin2 (b), we can average to get: sin2 (c) = sin2 (a)

[1 + cos2 (a)] [1 + cos2 (b)] + sin2 (b) . 2 2

More elegantly, sin2 (c) = sin2 (a) + sin2 (b) − sin2 (a) sin2 (b).

Alternative Pythagoras

Todhunter lists: c a b b a sin2 ( ) = sin2 ( ) cos2 ( ) + sin2 ( ) cos2 ( ). 2 2 2 2 2 Most importantly, for each of these, as r → ∞ we get 

|c| r

2

=



|a| r

2

+



yielding the classical Pythagorean: |c|2 = |a|2 + |b|2 .

|b| r

2

The world’s most famous right angle triangle Application: Can you name a right angle triangle on earth? London, North Pole, New Orleans! Given the location of both London and New Orleans, we can now work out the distance between them.

More generally, the Cosine Rule for Sides permits the evaluation of the distance between any two points on earth whose longitude and latitude are known, as we then have a SAS triangle situation.

The Sine Rule for Spheres ASA for small spherical triangles can be solved with the aid of the Sine Rule: sin(α) sin(β) sin(γ) = = . sin(a) sin(b) sin(c) As in planar case, all small spherical triangles can be analyzed in terms of right angle triangles. If ABC is a triangle and we drop a perpendicular from B to AC, where this new segment is denoted by d (CLICK: perp) then sin(α) =

sin(d) sin(c)

sin(γ) =

sin(d) sin(a)

so that eliminating d we get

sin(α) sin(a)

=

sin(γ) sin(c) .

Duality on a Sphere

Associated with each spherical triangle ABC is another triangle A’B’C’, the dual (or polar) of ABC, which is determined as follows: To find, e.g., A’, look at the two poles determined by the great circle through B and C, and chose the one which is on the same side of arc BC as A is. Repeat for the other vertices and sides. Check: the dual of the dual is the original triangle back! Is there a triangle which is its own dual? If so, can you characterize all such triangles?

Duality Is there any relationship between the angles and sides of a spherical triangle and the sides and angles of its dual? Basic fact of duality: the angles and sides of a spherical triangle are respectively the supplements of the sides and angles of its dual. In other words, α + a′ = β + b ′ = γ + c ′ = 180o and α′ + a = β ′ + b = γ ′ + c = 180o

Duality

Now we can use the Cosine Rule for Sides: cos(c) = cos(a) cos(b) + sin(a) sin(b) cos(γ) to derive a new (set of) formula(s) for solving ASA problems. What if we knew α, β and c, how could we detemine γ? Using duality, we easily get the Cosine Rule for Angles: cos(γ) = − cos(α) cos(β) + sin(α) sin(β) cos(c). Is this understandable/believable as the radius goes to infinity? Is this understandable/believable for right angle triangles?

Area on a sphere Another fun exploration is area. One might expect the easiest areas to find are rectangles (the basis of integral calculus . . . ) and circles. Not so! Even the area of a triangle is tricky to find: best expressed in terms of the vertex angles (in radians) rather than the sides. CLICK: Triangle area applet Girard’s Theorem: Area(triangle) = (α + β + γ − π)r 2 . Can lead students to discover this for themselves—often makes a big impression! α + β + γ − π is called the angle excess. CLICK: Girard applet

Area of a small spherical triangle

Consequences of Girard’s Theorem: ◮

The (vertex) angle sum for a spherical triangle always exceeds a half-turn.

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There are no perfect maps: any map (projection) of a sphere to a plane which sends great circles to lines must distort angles.

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Similar triangles do not exist: one can’t scale a given triangle up or down.

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Can’t tell so easily if our space is a 3-sphere.

Is Space Spherical? How could a bug tell if it was living on a 2-sphere of radius r ? It could follow a geodesic from point A to point B and see if it came back to A again. If this happened repeatedly, for different pairs of points, and the bug travelled the same distance each time, it could be argued that the bug was on the surface of a sphere, and r could also be determined. On the other hand, if any such trip did not take the bug back to its starting point, it would not be on the surface of a sphere. How can we tell if we are living in a 3-sphere in a higher dimersional space? Gauss allegedly measured the angles between three mountain tops in Germany, to see if the angle sum was 180o . Given the precision of the instruments of his day, it seemed to be . . .

The Shape of Space If we are living on a 3-sphere of radius r , then, given any three (non-collinear) points, they determine (a triangle on) a 2-sphere of the same radius r . Might we find a big enough triangle, out there in our solar system, and detect an angle sum in excess of 180o ? If so, while it wouldn’t prove that we were living on a 3-sphere, we’d know for sure that were not living in Euclidean space! Girard yields: angle excess =

triangle area r2

The biggest triangle area we could hope to measure would be less than the area of our solar system which is roughly 5 × 1019 square miles. On the other hand, the radius of the universe would have to exceed the diameter of our galaxy, which is roughly 6 × 1017 miles. Plugging into Girard we find that the angle excess is at most 10−16 radians—which is far too small to be reliably detectable!

Spherical Geometry Applications ◮

Navigation on (boats), near (planes) and about (space shuttle) spheres.

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Cartography (map making). Can’t have all of distance, direction and area correct. CLICK: Australian perpective? CLICK: Peters projection CLICK: Anti-Peters tirade

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Astronomy

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Projective geometry

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Computer graphics (rendering 3D worlds on 2D screens using 4D coordinates)

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Digital Hollywood (Pixar/Dreamworks films)

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Video games

Cosmology: The Shape of Space Is our world spherical (in a 3-sense)? What other possible shapes exist for the universe? If it’s not planar or spherical, might it be hyperbolic? Jeff Weeks, a freelance geometer and an expert on hyperbolic spaces, has calculated that our universe may indeed have a finite geometry with a hyperbolic radius of 18 billion light years. The WMAP satellite currently taking pictures of the early universe will hopefully provide evidence one way or other in the next few years, so that humanity may at last know the shape of existence itself. CLICK: Is our universe hyperbolic?

The Hyperbolic Plane Let’s start with the hyperbolic plane, a 2D non-Euclidean surface whose very existance was only conceived (albeit in a different context) circa 1825. Think of flat Euclidean 2-space as a tiling of regular hexagons which, at every point, looks flat. It has zero curvature.

The Hyperbolic Plane Think of a 2-sphere as a homogeneous surface which, at every point, looks like a nice smooth max (mountain top), orientated appropriately. Approximations of it can be modelled using a mix of hexagons and pentagons.

Has (constant) positive curvature, given by the reciprocal of the radius r .

The Hyperbolic Plane The hyperbolic “plane” can be thought of as a homogeneous surface which, at every point, looks like a nice smooth saddle point, orientated appropriately. Approximations of it can be modelled in several different ways with paper, e.g., using a mix of hexagons and septagons.

The Hyperbolic Plane Locally:

(note the edible form). Globally, the hyperbolic plane has (constant) negative curvature, given by the reciprocal of its “radius” ρ. Instead of folding back on itself, like a ball, it unfolds infinitely outwards. But can we experience hyperbolic geometry in our world?

The Hyperbolic Plane Yes—just ask Bill Thurston, David Henderson & Daina Taimina! It’s excessively curly. Like some lettuce, or kelp, or the frills or ruffles of sea slugs, flatworms and nudibranches. The hyperbolic plane can also be crocheted.

Curvature

Let’s compare the three basic geometries in a simplistic way:

What’s different about hyperbolic space? Geodesics are harder to visualize.

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Distinct geodesics need not meet at all. There is an abundance of “parallel” geodesics. (Given a line L and a point not on L, there are at least two distinct lines which contains the point, and do not meet L.)

Hyperbolic Geometry All the trig we discussed on the sphere works here if we replace the circular trig functions by hyperbolic ones. E.g., Pythagoras for hyperbolic right angle triangles says cosh(c) = cosh(a) cosh(b) when γ = 90o , where cosh(c) = 12 (e c + e −c ) is the hyperbolic cosine function. Whereas spherical triangles bulge at the corners, hyperbolic triangles have pinched corners. The vertex angle sum for triangles always falls short of 180o ! In fact, Drarig says: Area(triangle) = (π − (α + β + γ))ρ2 . Here π − (α + β + γ) is called the angle defect.

Hyperbolic Geometry Surprising consequence: there is a fixed upper bound on triangle areas. Infinitely big triangles have bounded area!

There are other popular models of the hyperbolic plane (e.g., the Poincare disc model, the upper half plane model etc), which we have not considered here. CLICK: Applet 1 for hyperbolic triangle area CLICK: Applet 2 for hyperbolic triangle area

Hyperbolic Applications ◮

Visualizing the structure of the World Wide Web in 3D hyperbolic space (Tamara Munzner et al, mid ’90s) “The felicitous property that hyperbolic space has ‘more room’ than Euclidean space allows more information to be seen amid less clutter, and motion by hyperbolic isometries provides for mathematically elegant navigation.” CLICK: Adaptive viewing of complex web graphs 1 CLICK: Adaptive viewing of complex web graphs 2

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Modelling the human brain CLICK: It isn’t brain surgery . . . yet!

Hyperbolic applications ◮

Mapping spheres with hyperbolic methods “One of the advantages of creating maps in the hyperbolic plane is that you can interactively change the map focus to alter to regions of high map distortion. Because we look at a hyperbolic map with Euclidean eyes, to us the center (or focus) of the hyperbolic map appears to have little distortion while the distortion increases as you move out to the edge of the map.” CLICK: Hyperbolic world maps

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Did we mention cosmology?

Flat Earth Society Membership Application Test Questions On flat surfaces, the shortest distance between two points is always a straight line, right? Okay, now solve this classic puzzle: In a room that is 12 x 12 x 30 (in feet), there is a spider near the top of one of the smaller walls. She is one unit foot from the ceiling, and centered between the side walls. She sees a fly on the opposite wall, one foot from the floor and centered between the side walls. What is the shortest path that she can travel (on walls, ceiling, and/or floor) to get to the fly (which remains fixed throughout)? Hint: it doesn’t matter that the units listed are feet and not metres! CLICK: Spider and fly Hint: put your calculator down, you will not need it.

Flat Earth Society Membership Application Test Questions CLICK: Spider and fly solution New version: In a room that is 12x12x24 (in feet), there is a spider at one corner. Where should the fly position himself so as to be as far as posssible from the spider? We seek the longest shortest path possible between the spider and the fly (along walls, ceiling, and/or floor) and the fly. Guess: the fly should sit it out at the diametrically opposite corner. CLICK: Spider and fly applet What two points on the surface of this room are the furthest apart anyway?

Shakespeare also weighed in on geometry

CLICK: 2D or not 2D

References David Henderson & Daina Taimina, Experiencing Geometry–Euclidean and Non-Euclidean with History, Prentice Hall, 3rd edition, 2005. George Jennings, Modern Geometry with Applications, Springer-Verlag, 1994. J.L. Berggren, Episodes in the Mathematics of Medieval Islam, Springer-Verlag, 1986. I. Todhunter, Spherical Trigonometry, Macmillan, 5th edition, 1886 (available from Cornell University Mathemetics Library Reformatted Books at: http://math.cornell.edu/∼ library/reformat.html) Jeffrey Weeks, The Shape of Space, Marcel Dekker, 2nd edition, 2002. What is the shape of the Universe? Google “StarChild Question of the Month for July 2001” (http://starchild.gsfc.nasa.gov/docs/StarChild/questions/question35.html)