Fundamental Tactics for Solving Problems : SYMMETRY

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SYMMETRY Many fundamental problem-solving tactics involve the search for order. Often problems are hard because they seem chaotic or disorderly; there appear to be missing parts (facts, variables, patterns) or the parts do not seem to be connected.

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SYMMETRY We all have an intuitive idea of symmetry; for example, everyone understands that circles are symmetrical. It is helpful, however, to define symmetry in a formal way, if only because this will expand our notion of it. We call an object symmetric if there are one or more non-trivial "actions" that leave the object unchanged. We call the actions that do this the symmetries of the object.LOGO

SYMMETRY SYMMETRY The most dramatic form of order . We call an object symmetric if there are one or more non-trivial "actions" that leave the object unchanged. We call the actions that do this the symmetries of the object.

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SYMMETRY Example 1.A square is symmetric with respect to reflection about a diagonal. The reflection is one of the several symmetries of the square. Other symmetries include rotation clockwise by 90 degrees and reflection about a line joining the midpoints of two opposite sides.

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SYMMETRY A circle has infinitely many symmetries, for example, rotation clockwise by a degrees for any a.

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SYMMETRY . The doubly infinite sequence.... ,3,1,4,3,1,4,3,1,4, ... is symmetrical with respect to the action "shift everything three places to the right (or left)."

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SYMMETRY Why SYMMETRY? Because it gives you "free" information. If youknow that something is, say, symmetric with respect to 90-degree rotation about some point, then you only need to look at one-quarter of the object. And you also know that the center of rotation is a "special" point, worthy of close investigation.

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SYMMETRY two things to keep in mind as you ponder symmetry:1.The strategic principles of peripheral vision and rule-breaking tell us to look for symmetry in unlikely places, and not to worry if something is almost, but not quite symmetrical. In these cases, it is wise to proceed as if symmetry is present, since we will probably learn something useful.

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SYMMETRY2. An informal alternate definition of symmetry is "harmony." This is even vaguer than our "formal" definition, but it is not without value. Look for harmony, and beauty, whenever you investigate a problem. If you can do something that makes things more harmonious or more beautiful, even if you have no idea how to define these two terms, then you are often on the right track.

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SYMMETRY 2 ways of solving Symmetry: Geometric Algebraic

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A square is inscribed in a circle that is inscribed in a square. Find the ratio of the areas of the two squares.

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SYMMETRY Algebraic SymmetryDon't restrict your notions of symmetry to physical or geometric objects. For example, sequences in Pascal's Triangle: 1,6,15,20,20,15,6,1. That's only the beginning. In just about any situation where you can imagine "pairing" things up, you can think about symmetry. And thinking about symmetry almost always pays off.LOGO

SYMMETRY 1. Prove that: n (n + 1) 2 (n + 2) 2 + (n + 3) 2= 4 Then what is the value of n?

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SYMMETRY 2. Prove that ( a + b) ( b + c) ( c + a) 8abc is true for all positive numbers a, b and c, with equality only if a = b = c.

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3. Find the product (a k)(a)(a + k)(a + 2k)

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SYMMETRY 4. Find the axis of symmetry in the equation y=x2-4x+3

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SYMMETRY 5. Find the length of the shortest path from the point (3,5) to the point (8,2) that touches the x-axis and also touches the yaxis.

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SYMMETRY

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SYMMETRY Factor a3+ b3 + c3 - 3abc.

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SYMMETRY.Recall that the parabola is defined to be the locus of all points in the plane, such that the distance to a fixed point (the focus) is equal to the distance to a fixed line (the directrix). Prove the reflection property of the parabola: if a beam of light, traveling perpendicular to the directrix, strikes any point on the concave side of a parabolic mirror, the beam will reflect off the mirror and travel straight to the focus.

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