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I ♥ Parabola - Poster
PARABOLA Bridges are physical manifestations of a parabola. The height-versustime graph of falling objects at or near the center of the earth can be modeled by a parabola as well. It is usually the first nonlinear curve that students study in algebra. A parabola can be defined as the locus of points equidistant from a point (called the focus) and a line (called the directrix). The parabola with vertex at the origin and opening upward has the equation 4 py = x2, where p is a constant representing the distance from the focus to the vertex (or the vertex to the directrix).
I ♥ Fibonacci Numbers - Poster
FIBONACCI NUMBERS The Fibonacci sequence, 1, 1, 2, 3, 5, 8, 13, 21, 34, . . . , where the first two terms in the sequence are each 1 and every successive term is the sum of the two previous terms.
The ratio of successive terms approaches the golden ratio, ø = (1 + √5) / 2 ≈ 1.6803399. Fibonacci numbers occur often in nature, in particular in phyllotaxy, the study of leaf arrangement on a stem. The spirals on sunflower seed heads, on pineapples, and in pine cones are invariably Fibonacci numbers or multiples of Fibonacci numbers.
I ♥ Truncated Icosahedrons - Poster
TRUNCATED ICOSAHEDRON An icosahedron is one of the five Platonic solids (tetrahedron, cube, octahedron, and dodecadhedron are the others). The 20 faces of the icosahedron are congruent equilateral triangles. If planes sliced the icosahedrons at the 1/3 point of each edge, each triangle would become a hexagon, and a pentagon would replace each vertex of the original icosahedron.
The soccer ball and the ball used in the Olympic sport of team handball is an example of a spherical rounded analog of a truncated icosahedron. The flexibility of the surface of these balls and the air pressure on their contents results in this familiar round shape.
I ♥ Logarithmic Spirals - Poster
LOGARITHMIC SPIRAL The logarithmic spiral is a polar curve whose equation is given by r = a e , where bθ a and b are constants.
The nautilus shell is the best-known example of the logarithmic spiral found in nature, but flying animals will approach targets by using a logarithmic spiral pattern. Raptors will hawk prey, swifts will attack insects, and insects will approach a light source via a logarithmic spiral. Beaches can form in the shape of a logarithmic spiral, and the arms of tropical storms can also model this shape. The logarithmic spiral has close connections to the golden ratio.
I ♥ Tori - Poster
TORI Doughnuts, inner tubes, and bagels are all examples of tori (singular, torus)
found in the real world.
A torus can be thought of as the shape formed when a circle is rotated 360˚ around a line in its plane that shares no points in common with the circle. Thus the circle with equation ( x – 2) 2 + y 2 = 1 will create a torus if it is rotated around the y-axis.
I ♥ Paraboloids - Poster
PARABOLOID When a parabola is rotated about its axis of symmetry, the three-dimensional surface thus created is a paraboloid. The general equation of a paraboloid in a three-dimensional space is
z=( x 2/ a 2)±( y 2/ b 2)
where a and b are constants. Elliptic paraboloids (in which the terms are added) are used for satellite dishes and in the reflectors in automobile headlights because of the reflective property of a parabola; Pringles potato chips are examples of hyperbolic paraboloids (in which the terms are subtracted).
I ♥ Hyperbolic Cosines - Poster
HYPERBOLIC COSINE Sometimes called the catenary, the hyperbolic cosine is the function whose equation is
=cosh( y )=( x + e x )/2. e - x
The shape of the graph of this function is similar, but not identical, to that of a parabola. To see the difference, students can use their graphing calculators to try to superimpose = cosh ( y x) on the parabola y = x 2 + 1.
While suspension bridges often take the shape of a parabola, any bridge suspended only from its endpoints with uniform weight throughout will take the shape of a catenary or hyperbolic cosine. The Gateway Arch in St. Louis is an inverted catenary slightly flattened at the top, and Antonio Gaudi and other architects have used inverted catenary arches in their buildings.
18" x 24" inches.
I ♥ Oblate Spheroids - Poster
Oblate spheroids are bodies that are shaped like a sphere but are not perfectly round, particularly an ellipsoid, which is generated by revolving an ellipse around one of its axes.
I ♥ Conical Frustums - Poster
I ♥ Conical Frustums - Poster
A conical frustum is formed by slicing the top off a cone. The frustum is what is left. The word frustum comes from the Latin expression for piece or bit.
Fibonacci Stays Here Poster
Fibonacci stays here
is also referred to as the golden section, the golden mean, the golden proportion, the extreme and mean ratio, and the divine proportion. It has been used to ensure beauty and balance in art, architecture, music and design for centuries. THE GOLDEN RATIO
Here, the ubiquitous Fibonacci sequence shows up where it is least expected: in the logo of a Chicago hotel. Your students can discover how to construct and measure a golden rectangle, determine where the golden ratio appears in the pentagon formed by a 5-pointed starfish, and apply graphs of polar equations. (Solutions provided.)