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 The study of mathematics is the study of patterns. Look around you! There are patterns everywhere. Watch a sunset, or clouds blowing by, or the leaves on a tree. Listen to music, or language, or a bird song. It is through patterns that we can come to understand our lives and the world we inhabit. In mathematics, when you can see the patterns in numbers, in shapes, in functions--then you can begin to understand the mathematics.
 And everywhere you look, there are doors. Some doors are ancient, and do not seem to lead where you think you are going. Others are new and bright, and seem to promise whatever you can imagine. This is what is important about doors: Behind every door is a door. Think of it as just another definition of infinity. What if it turns out that success depends not just on which doors you pass through, but on how many doors you pass through?

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