*Note: This is an abbreviated Project Idea, without notes to start your background research, a specific list of materials, or a procedure for how to do the experiment. You can identify abbreviated Project Ideas by the asterisk at the end of the title. If you want a Project Idea with full instructions, please pick one without an asterisk.
What do knots, maps, mazes, driving directions, and doughnuts have in common? The answer is topology, a branch of mathematics that studies the spatial properties and connections of an object. Topology has sometimes been called rubber-sheet geometry because it does not distinguish between a circle and a square (a circle made out of a rubber band can be stretched into a square) but does distinguish between a circle and a figure eight (you cannot stretch a figure eight into a circle without tearing) (Wikipedia contributors, 2006). A common joke is that topologists are people who don't know the difference between a coffee cup and a doughnut. A project in topology can have many forms. Can "Euler's Solution" help you efficiently run your errands? Can you figure out the number of possible routes to get to school? What is the minimum number of colors needed to color in a U.S. map so that no two states that are touching have the same color? Use topology to untangle knots or to discover knots that cannot be untangled. Use topology to solve mazes, draw circuit diagrams, make phylogenetic trees, or fold origami! The possibilities are endless... (Britton, 2006)
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If you like this project, you might enjoy exploring these related careers:
Cartographers & Photogrammetrists
Maps can give us much more information than ways to get from A to B. Maps can give us topographic, climate, and even political information. Cartographers and photogrammetrists collect a vast amount of data, such as aerial data and survey data to produce accurate maps and models. For example, by collecting rainfall data, a cartographer can make an accurate model of how rainfall can affect an area's watershed. The maps and models can then be used by policy makers to make informed decisions.
When you hear the word geography, you might think of maps and names of state capitals, but the work of geographers is much more than creating maps and identifying places. Geographers look at how people, places, and Earth are connected. They study the economy, social conditions, climate, and topography of a region to help answer questions in urban and regional planning, business, agriculture, and medicine.
Essential members of any construction team include mapping and surveying technicians—the "instrument people"—who set up and operate special equipment that measures distances, curves, elevations, and angles between points on Earth’s surface. These technicians then take the data gathered by the instruments and create maps and charts on a computer. About half of their work is spent in hands-on, high-technology data collection in the field, while the other half is spent in an office—they get to experience both worlds and create documents that define, in great detail, places on Earth.
Mathematicians are part of an ancient tradition of searching for patterns, conjecturing, and figuring out truths based on rigorous deduction. Some mathematicians focus on purely theoretical problems, with no obvious or immediate applications, except to advance our understanding of mathematics, while others focus on applied mathematics, where they try to solve problems in economics, business, science, physics, or engineering.
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