Term
How many cameras are needed to ensure that every point in a gallery
is seen by some cameras? |
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Definition
| The # of cameras needed is dependent upon the # of vertices in the gallery. You would then divide the # of vertices by 3 in order to figure out the number of cameras you would need. |
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Term
| What does the Art Gallery Theorem say? |
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Definition
| Given polygonal closed curve in the plane with v vertices, there are v/3 vertices from which it is possible to view every point on the interior of the curve. If v/3 is not an integer , then the largest # of vertices is the biggest integer less than v/3. v= # of vertices. |
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Term
| What is the main idea of the proof of the Art Gallery Theorem ? |
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Definition
| Any triangle has 3 vertices and from each vertex it is possible to view every point in the triangle. The key is to divide it into a whole bunch of triangles. That is, convert the complex data into several easy ones. |
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Term
If a gallery has a convex shape how many cameras are needed to ensure
that every point in a gallery is seen by some cameras? |
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Definition
|
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Term
| What is a Golden Rectangle? |
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Definition
| Any rectangle having base b and height h such that b/h=1.618... or phi |
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Term
| What is the Golden Ratio? |
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Definition
| If we divide the length of the longer side by the length of the shorter side of a golden rectangle, we get the golden ratio or 1.62, or phi. |
|
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Term
| Where does the Golden Ratio appear? |
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Definition
| Pinecones, pineapples, daisies, golden rectangles. |
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Term
| How can you create a Golden Rectangle? |
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Definition
| make the length of the base relative to the length of the height. Start with a square and elongating it by using a simple geometric procedure. |
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Term
What does it mean that the Golden Rectangle has another within a Golden
Rectangle? |
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Definition
| If a golden rectangle is divided into a square and a smaller rectangle, then the smaller rectangle is another golden rectangle |
|
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Term
| What does dimension mean? |
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Definition
| In physics and mathematics, the dimension of a space or object is the minimum number of coordinates needed to specify any point within it. |
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Term
| What is the fourth dimension? |
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Definition
| The three dimensions we already know and love--(Length, height, width)--plus another. (time//a “new direction”) ---someone feel free |
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Term
| Can you think of any examples of the fourth dimension? |
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Definition
| hypercube or a tesseract [from either transformers or a wrinkle in time] |
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Term
| How would 4D creatures see us? |
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Definition
| They would be able to see our internal organs; as flatlanders basically. |
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Term
| What does it mean that two objects are equivalent by distortion? |
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Definition
| Two objects are equivalent by distortion if we can stretch, shrink, bend, or twist one, without cutting or gluing, and deform it into the other. |
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Term
| Can you give some examples of objects that are equivalent by distortion? |
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Definition
| Triangle, square and pentagon are all equal by distortion. |
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Term
| Can you give some examples of objects that are not equivalent by distortion? |
|
Definition
|
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Term
| Is the torus equivalent by distortion to the sphere? Why or why not? |
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Definition
| If you cut into the torus, only one section remains- basically a hollow tube. However, if you cut into a sphere, two sections will be left over, proving that they are not equivalent by distortion. |
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Term
| What is the Mobius Band? How do you make a Mobius band? |
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Definition
| A Möbius band is surface with one side and one edge modeled as follows: take a strip of paper and give it a half twist. |
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Term
| How many edges and sides does a Mobius Band have? |
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Definition
| Just one on both, the twist makes it seem like two- illusion. |
|
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Term
| How does the Mobius Band compare with the cylinder? |
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Definition
| You would have to cross an edge on the cylinder to get to the inner part, whereas on the strip no crossing is required, you would eventually reach that point due to the half twist. |
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Term
| What happens if you cut the Mobius Band along its circumference? |
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Definition
| As shown on the third experiment in page 346, if you cut along the right edge you would end up with a new band made up of two edges. |
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Term
| When would you obtain a double twisted cylinder starting with a Mobius Band? |
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Definition
|
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Term
| What is the Klein Bottle? |
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Definition
| A one sided 3 dimensional surface which passes through itself; its insides are the same as the outside. |
|
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Term
| How many faces does a Klein Bottle have? |
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Definition
|
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Term
| What is Konigsberg Bridge Challenge? |
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Definition
| The question of whether we can walk over a collection of bridges that connect various landmasses without going over the same bridge twice. |
|
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Term
| Can one solve the Konigsberg Bridge Challenge? |
|
Definition
|
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Term
| What is a graph? What is a connected graph? |
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Definition
A graph is an object drawn in the plane using a finite set of points, or vertices, with line segments or curves, called edges, connecting pairs of vertices.
A graph is connected if for any two vertices there is an edge (path) that connects them. |
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Term
| What is an Euler Circuit? Can you write an example? |
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Definition
| An Euler Circuit is a path that traverses each edge of a graph exactly once, reaches every vertex and returns to the starting vertex. |
|
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Term
What does the Eulers Circuit Theorem say? What is the link between
the Eulers Circuit Theorem and the Konigsberg Bridge Challenge? |
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Definition
Euler’s Circuit Theorem (describes all graphs that have an Euler Circuit)
A connected graph has an Euler Circuit if and only if Every Vertex is even.
Otherwise, the Konigsberg challenge is impossible. |
|
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Term
If a graph has no Euler Circuit (or it is not an Euler Circuit), would
you be able to add edges to make it an Euler Circuit? |
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Definition
|
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Term
What is the minimum number of colors that always suffice to color any
potential world map? |
|
Definition
|
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Term
| What does the Four Color Theorem say? |
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Definition
| No more than four colors are required to color the regions of the map so that no two adjacent regions have the same color. Two regions are called adjacent if they share a common boundary that is not a corner, where corners are the points shared by three or more regions. |
|
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Term
| What does the Six Color Theorem say? |
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Definition
| Given a map consisting of connected regions printed on a sheet of paper the regions can be colored using only six different colors such that any two different regions sharing a common border will have different colors. |
|
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Term
| What is the main idea of the Proof of the Six Color Theorem? |
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Definition
| The proof is a proof by contradiction, assume that color theory is not true. Thus there would be a map that had more than 6 colors. Actual proof: (Facts) E is less than 3V-6 for any planar graph with no loops or multiple edges having a vertex count of V (V at least 3). (Proof by contradiction) Suppose you could draw a graph where each vertex has degree 6. Then since each edge has two ends, the number of edges E in that graph would be E=6V/2=3V. But this is impossible. If you had a planar graph then realize there is always going to be a vertex with degree 5 or less than five connections. |
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Term
| What are some applications of the Coloring Theorems? |
|
Definition
| GPS, 3D models, city planning blueprints |
|
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Term
| What is the meaning of a Network? |
|
Definition
| A graph of connections between objects/people/etc... |
|
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Term
| Can you give some examples of Networks? |
|
Definition
| Facebook, UT Alumns, Ronald Reagan supporters. |
|
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Term
| What is the meaning of Degrees of Separation? |
|
Definition
| A degree of separation is a measure of social distance between people. You are one degree away from everyone you know, two degrees away from everyone they know, and so on. |
|
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Term
| Is there such a thing as the Six Degrees of Separation Theorem ? |
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Definition
| Not that I know of other than the actual definition of Six degrees of separation. |
|
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Term
|
Definition
| A self repeating pattern that builds inwards. It looks the same or similar even when looking at a microscopic piece of it. |
|
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Term
| What is the Sierpinski Triangle? |
|
Definition
| Self repeating triangle: At stage 2, the Sierpinski triangle has 9 filled in triangles. At stage 3 there are 27. At each stage, the number of triangles increases by a factor of 3. At stage 4 there are 81 triangles. These values suggest that at stage n there will be 3n triangles. |
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Term
| How can you turn a triangle into a snowflake? |
|
Definition
1.
divide the line segment into three segments of equal length.
2.
draw an equilateral triangle that has the middle segment from step 1 as its base and points outward.
3.
remove the line segment that is the base of the triangle from step 2. |
|
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Term
| What does Self-Similarity mean? |
|
Definition
| That it is composed internally of the same sequence, i.e. the above koch snowflake. All the smaller segments extending from the original piece are smaller copies. |
|
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Term
| What is the Kochs Kinky Curve? |
|
Definition
| One example of an infinite replacement process, similar to the Koch Snowflake. |
|
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Term
| How is the Sierpinski Triangle generated ? |
|
Definition
| Start with a filled in triangle and replace it with an upside down triangle, hollowing out the original space. Continue this process in due scale with each remnant triangle.... page 471 |
|
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Term
How can you generate fractals? What is a process of repeated replacement?
What is the collage method ? |
|
Definition
| Following a set of collage-making instructions will lead to the same image no matter what picture we start with. |
|
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Term
| What is the final image when one applies the Sierpinski Triangle process to any image? |
|
Definition
| Exactly another Sierpinski Triangle! |
|
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Term
| What are some examples of Dynamical Systems? |
|
Definition
| Population Shifts, Weather, planets, stars, temperature. |
|
|
Term
| What is the Game of Life? |
|
Definition
1- A living square will remain alive in the next generation if exactly 2 or 3 of the adjoining 8 squares are alive in this generation.
2- A dead square will come back if 3 of its adjoining squares are alive. |
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Term
| What are some examples of population that correspond to a periodic, stable, migratory or an extinction for the Game of Life? |
|
Definition
| periodic: bacteria, Migratory: Birds, Extinction: overpopulation of fish in a pond |
|
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Term
| What is the Verhulst Model? How do you compute fish population density? |
|
Definition
If the population is far from the maximum sustainable population, expect a large increase. If the population is not as far from the maximum sustainable population, we expect a lesser increase. Finally, if the population exceeds the maximum sustainable population, expect a decrease.
Example: Maximum sustainable population for a pond is 5000. If in year 2 the fish population of the pond was 2500 and year three 3000, population densities for year 2 would = 2500/5000=0.5 and for the third year 3000/5000=0.6.
The next step would be to take the change and divide it by the initial population density of 0.5, so 0.1/0.5= 0.2, or the rate at which the population increased. In other words, the population increased by 20% from years 2 to 3. |
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Term
If you deposit $1, 000 in a savings account that pays 10 percent interest
compounded annually, which is the best estimate for the amount of time in years required to double your money? |
|
Definition
| 7.2 years. Rule of 72 is that you can find the amount something doubles by dividing the interest rate by 72. 72/10=7.2 years. You can also use natural logs and you will come up with the same answer. |
|
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Term
| What is the rule of 72? What is the rule of 70? |
|
Definition
| For instance, if you were to invest $100 with compounding interest at a rate of 9% per annum, the rule of 72 gives 72/9 = 8 years required for the investment to be worth $200. |
|
|
Term
| Find a formula that explains how many years it will take for your balance $X to become $1.5X if you deposit $X in a savings account that pays 5 percent interest compounded annually |
|
Definition
|
|
Term
| What is Chaos (Chaos Theory)? |
|
Definition
That some variables are unmeasurable...
The idea that it’s impossible to predict some aspects of the future, as shown by minor rounding differences between two calculators creating very different results after several stages of calculating. |
|
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Term
If you use two different calculators to do the same calculations, are you
guaranteed to always get exactly the same answer? Why or why not? |
|
Definition
| We are not always guaranteed the same answer- different manufacturers may program them differently, and the answer also depends on the number of decimal points or available spaces on the display. |
|
|
Term
| What is the Butterfly Effect? |
|
Definition
| One small action-such as the flap of a butterfly’s wing-could offset much bigger actions, such as weather patterns |
|
|
Term
| Can you list some examples that relate to Chaos? |
|
Definition
| traffic, health pandemics, market prices |
|
|
Term
| Can we accurately predict Planetary Positions for millions of years? |
|
Definition
| No, calculations will begin to differ from year to year; the more number of years, the more room for differences |
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