Frieze patterns are patterns that repeat in a straight vertical or horizontal line. Frieze patterns are found in architecture, fabrics, and wallpaper borders. As we learned in class, there are only seven types of frieze patterns.
As I was looking for frieze patterns around the house, I found that the headboard of my bed has a frieze pattern. Using the reflection chart, I figured out that this frieze pattern is “F1”. It has no vertical or horizontal reflection and it also has no half turn symmetry.
Optical illusions are really cool and fascinating. As I was doing research on optical illusions, I came across this amazing illusion called the Checker Shadow Illusion.
The Checker Shadow Illusion was created in 1955 by Edward H. Adelson, professor of Vision Science at MIT.
In this visual illusion, the square marked with an A looks a lot darker than B, right? But in reality, they are the same shade of grey.
Explanation of the Illusion:
Your brain makes a lot of assumptions when it’s processing visual information. The visual system uses several tricks to determine where the shadows are and how to compensate for them, in order to determine the shade of grey “paint” that belongs to the surface.
The first trick is based on local contrast. In shadow or not, a check that is lighter than its neighboring checks is probably lighter than average, and vice versa. In the figure above, the light check in shadow is surrounded by darker checks. Thus, even though the check is physically dark, it is light when compared to its neighbors. The dark checks outside the shadow are surrounded by lighter checks, so they look dark by comparison.
A second trick is based on the fact that shadows often have soft edges, while paint boundaries (like the checks) often have sharp edges. The visual system tends to ignore gradual changes in light level, so that it can determine the color of the surfaces without being misled by shadows. In this figure, the shadow looks like a shadow, both because it is fuzzy and because the shadow casting object is visible.
Part of the problem is where the squares are located. Most people know enough about checker or chessboards to know that a square will be the opposite color of all the adjacent squares. That means two squares that are next to each other, or separated by an even number of squares, should be different colors. Since squares A and B are two squares away, they logically should be different colors, and our brains really seem to want to think that as well.
Proof of the Illusion
The Checker Shadow Illusion In Real Life
The Checker Shadow illusion is one of the most stunning illusions, as it lets viewers perceive all the boxes as a different color when all of them are the same.
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As we learned in class, the goal of a traveling salesman problem (TSP) is that a salesman should travel to every single city in an area, visiting each city only once. The salesman also needs to end up in the same city where he starts his journey.
We also learned about Hamilton’s path and circuit. Similar to Euler’s path, a Hamilton path is one that passes through each vertex exactly once. A Hamilton circuit is actually just a Hamilton path, but is a cycle.
In 1857, William Rowan Hamilton developed a mathematical game called the Icosian game. The game’s object is finding a path along the edges of a dodecahedron such that every vertex is visited exactly once, and the ending point is the same as the starting point. Hamilton’s game was eventually turned into an actual board game. And, if we think about it, even the Icosian game is really just another variation on the traveling salesman problem! Just as the dodecahedron has 20 vertices, we could imagine Hamilton’s game as a map of 20 cities. Using this analogy, our traveling salesman would need to find a way through 20 cities, visiting each city once, and ending up at the starting point.
Solution To The Icosian Game in 2D and 3D:
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Voting is used in formal and informal settings, such as political elections, electing officers to various positions or even when a group of people is deciding which restaurant to go to. I found the instant runoff method to be the most interesting because the voter has the ability to choose all the candidates they like instead of just choosing one candidate.
Instant Runoff method is a ranked choice voting system in which each voter ranks the list of candidates in order of preference. These ranking are used to elect a majority winner in a single election. This voting method eliminates voters from choosing only one candidate, since voters can vote for their favorite candidate, knowing that if their first choice does not win, their vote will not be wasted. There are many benefits of using instant runoff method such as it saves money, increases voter participation, and it also prevents the hassle of running another election.
Did you know some cities in California use the instant runoff method to elect its mayors, city attorney, and city council? San Francisco has been using this method since past four years and it turns out that voter participation has increased tremendously. Using instant runoff method has also helped save money. Other cities in California such as Oakland, Berkeley, and Santa Clara also use instant runoff method to elect candidates.
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