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Pythagoras
Euclid
Archimedes
The Golden Mean or Rectangle
Pictures of things related to the Golden Rectangle
Pythagoras of Samos
Pythagoras was born in about 569 BC and died about 479 BC He greatly contributed to the study of mathematics. In fact, had he never been born, many math concepts and theories would never have existed, such as the Golden Rectangle. Even though it is unknown who exactly discovered the Golden Rectangle, Pythagoras contributed greatly to it. Perhaps one of his more famous discoveries is the Pythagorean Theorem. The Pythagorean Theorem is a formula used to determine the hypotenuse of a right triangle using the two side lengths. (a squared plus b squared is equal to c squared) Pythagoras also was a great musician, writer, and philosopher.
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Euclid
Around 325 BC, Euclid was born in Alexandria. He is famous for writing a series of books called The Elements. These books are about all of his mathematical teachings. Euclid made some of the most important advancements in math, including many properties about geometrical shapes and formulas for algebra. The books were so effective that even today his books are still used in teaching high school students about math.
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To see samples of his books, click here.
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Archimedes
(287 BC- 212 BC)
Archimedes, born in Syracuse, Sicily, was known for his advancements in science as well as math. Perhaps one of his most famous discoveries was Archimedes' Principle. Archimedes' Principle states that any object, wholly or partly immersed in a fluid, is buoyed up by a force equal to the weight of the fluid displaced by the object. Basically, this says that when something is in water, it tends to float because of the force of the water pushing back on the object. This is the law that explains why ships float on water.
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To learn more about Archimedes' Principle, click here.
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The Golden Rectangle
The Golden Rectangle is a ratio that can be applied to just about everything in nature. Paintings, seashells, statues, even architecture has this ratio behind it. The ratio is about 1:1.618, or a little bit over half way. This was discovered by Pythagoras when he was studying a picture of a pentagram, as shown above. He noticed that if he stood the sides of the inner star upward, each one was about a 1:1.618 ratio. Pythagoras then drew a rectangle with another rectangle inside it that had the same ratio (shown below). He continued this pattern and realized that it went on forever. This is why it is often called The Golden Rectangle. Today, this ratio is used many times when constructing new buildings, since it has a pleasing effect on the eyes. To see some things involving the Golden Ratio, or Golden Proportion, click here.
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The Golden Rectangle is shown above. Inside of the big rectangle is another rectangle that has an exact ratio of 1:1.618. Inside of that rectangle that is also a perfect ratio of 1:1.618. More rectangles can be added on forever. Notice that if you draw a curved line from the bottom left vertex of the outer rectangle to the top right vertex of the outer rectangle and then to the bottom right vertex of the first inner rectangle and so on, you will get a spiral that continues to approach the center, but in theory will never touch itself. This spiral can be compared to the spiral of a seashell. This information is just touching up on the Golden Rectangle, but if you want to learn more, click here to go to a list of links to other places in this website.
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Pictures of Things Involving the Golden Ratio
This is a photocopy of the famous painting, the Mona Lisa. The Golden Ratio is applied to the face. Notice how from the bottom of her chin to the top of her eyebrows is just a little bit over half way up the Golden Rectangle. Same thing goes with the top of her forehead to the bottom of her nose. This is not unusual, since the ratio is where the eye naturally tends to rest.
Even the Parthenon uses the Golden Ratio. From the front view, there are over 4 Golden Rectangles used in the columns.
Notice how the shape of the seashell is very closely related to the shape of the spiral formed from the Golden Rectangle. Is this just a coincidence? Or is this the way nature intended everything to be?
For more information on the Golden Ratio:
Phi: That Golden Number
Golden Ratio
Fibonacci in Nature
The Golden Ratio
Math & Art: The Golden Rectangle
35mm Film and the Golden Rectangle
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