## Archive for the ‘Geometry’ Category

### Aperiodic tilings

March 20, 2012

This week, we will be looking at APERIODIC TILINGS! Most tiles can cover the plane (or your floor) in simple ways that repeat themselves. Led by our very own Dr. Clair, we’ll investigate some tiles (and sets of tiles) that tile without repetitions, such as the Penrose kite and dart tiles.

Join us in the Ritter Hall lobby on Wednesday from 4-5! Refreshments will be provided!

### Catenary curve finished

February 23, 2012

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### Catenary curves continued!

February 22, 2012

Last week, we started construction on a catenary curve. We were pretty close to finishing by the end of the meeting, but there is still some work to be done. Come help us put the finishing touches on our cardboard masterpiece tomorrow in the lobby of Ritter Hall from 4pm – 5pm! See you there!

### Catenary curves

February 14, 2012

This Wednesday from 4-5pm, we will look at catenary curves, like our own Gateway Arch. The equation for a catenary curve is of the form

$y = a\cosh(\dfrac{x}{a})$

But they look so much like parabolas! How are these curves like parabolas? How are they different? Come find out about all sorts of these types of curves in the lobby of Ritter Hall from 4-5pm tomorrow! Refreshments will be provided. Bring your friends!

### Pythagorean Triples

November 21, 2011

A Pythagorean triple is a triplet $(a,b,c)$, where $a, b, c$ are integers such that $a^2+b^2=c^2$.

It is called a Pythagorean triple because it can represent 3 sides of a right triangle. You probably already know two Pythagorean triples: $(3,4,5)$ and $(5,12,13)$. Are there other triples? How do we find them?

Let’s consider the equation
$a^2+b^2=c^2$
where $a, b, c$ are integers. If $a, b, c$ have a common factor $d$, say $a=da_1$, $b=db_1$, and $c=dc_1$. Then we have
$d^2a_1^2+d^2b_1^2=d^2c_1^2$
which is true if and only if
$a_1^2+b_1^2=c_1^2.$
So let’s assume that $a, b, c$ have no common factors. Such triples will be called primitive. Since odd perfect squares are congruent to 1 mod 4 and even squares are congruent to 0 mod 4, then $c$ must be odd and exactly one of $a$ or $b$ is even. Let’s suppose that $b$ is even: $b=2k$ for some integer $k$. Then
$4k^2=b^2=c^2-a^2=(c+a)(c-a).$
So both $c + a$ and $c - a$ must be even. Say $c + a = 2r$ and $c - a = 2s$. Then $rs=k^2$ and we have $c = r + s$ and $a = r - s$. Since $a$ and $c$ have no common factors, we must have that $r$ and $s$ have no common factors. Since $rs$ is a square, then both $r$ and $s$ are squares. Say $r=m^2$ and $s=n^2$. So $a=m^2-n^2, b=2mn, c=m^2+n^2$ if (a,b,c) is a primitive triple.

Hence, any Pythagorean triple, with $b$ even, is either of the form $(m^2-n^2,2mn,m^2+n^2)$ or it is a multiple of that form. You can plug in different values for $m$ and $n$ to get different right triangles with integer sides. Fun!

### 4-dimensional Hypercubes

October 24, 2011

The picture above is a drawing of a 4-dimensional hypercube. In general, we can define an n-dimensional hypercube for n a non-negative integer. The first 4 of these are

0: A point
1: A line segment
2: A square
3: A cube

We construct higher dimensional hypercubes from lower dimensional ones. We start with a 0-dimensional hypercube: a point.

Start with a point A and draw a line to a different point B. Let’s say the distance between A and B is d. This is a 1-dimensional hypercube: a line.

Now draw a line of length d, orthogonal to AB, from B to a point C, and a line of length d, orthogonal to AB, from A to a point D. This is a 2-dimensional hypercube: a square ABCD.

Now draw lines of length d respectively from each corner of the square ABCD, orthogonal to both AB and BC. Call the terminal points E,F,G,H. This is a 3-dimensional hypercube: a cube ABCDEFGH.

And so on for higher dimensions.

Drawing these on a piece of paper, a 2-dimensional plane, becomes more difficult as n increases. For example, a picture of a 7-dimensional hypercube looks like

How do we draw these pictures? In the 4-dimensional case, we can draw a hypercube from these steps. To get a better idea of the hypercube, we can watch the hypercube rotate

or rotate it ourselves.