arithmetic progression) and geometrics sequences (i.e. Patterns within mathematical sequences provide the key that reveals a common thread of how each number is connected to one another.īroadly speaking, mathematical sequences can be categorized into two major groups: arithmetic sequences (i.e. Each number within a mathematical sequence is identified as a term. Simply described, a math sequence is a group of numbers that follow a specific pattern. A more relevant memory today might be one of you reciting your times table. One’s earliest recollection of a math sequence probably began at the age of two, when you started counting to ten. What I've heard called a golden triangle is more like a golden rectangle.Math sequences can be discovered in your everyday life. I am positive it was in my textbook from Algebra 1. I have talked to my Geometry teacher about whether there is such a thing as a Golden Triangle, and he says no, but my Algebra Honors teacher tells me that there is, and it is a right triangle with the ratio 3:4:5, and any of the multiples, like if you multiply by 2, 6:8:10. More on the golden triangleĪ student in 2001 had a wrong impression of the meaning of the name: Golden Triangle: An Isosceles Triangle And this connection to the pentagon may be the main reason Euclid found this ratio worth exploring. This is true simply because the angles add up right. When we attach to AC and BC two triangles that are congruent to triangle ACD, we find a regular pentagon. Here is a pentagram (regular five-pointed star) showing the five golden triangles it contains: So BC : AB is this famous ratio that's why this triangle is called a Golden Triangle.īut there’s more! As an example of the appearance of Golden Triangles: the outside triangles of a pentagram are Golden Triangles. The number 1/2 + sqrt(5)/2 is known as the Golden Ratio, or Golden Mean. Suppose we have a triangle ABC, such that BD, in this case we must have x = 1/2 + sqrt(5)/2. Please help me find out what the Golden Triangle is. It has to do with math but so far all my results have something to do with bicycles. I have gone to many different Web sites trying to figure out what the Golden Triangle is, but I cannot find it. I'm doing a project for math where I have to answer questions using the Internet. Here is a question from 1999 about the Golden Triangle: Golden Triangle: What is It? It gets confusing after a while, doesn’t it? The Golden Triangle Together, these are used in the formula for the Fibonacci sequence:
But as I said, it doesn’t make much difference.
I’ve been using the upper case (\(\Phi\)) and lower case (\(\phi\)) forms the other way around, which is perhaps more standard. Since they are reciprocals, either could just as well be given that name. This equation (equivalent to x^2 - x - 1 = 0) is satisfied by both Phi and -phi, which therefore can be called the _golden ratios_. The latter facts together give the definition of the golden ratio: I answered, going along with his terminology: It's more than just coincidence: the golden ratio (as you define it) is phi's twin, "Phi," where
This student has evidently seen the first number in connection with geometry and architecture, and the second in connection with nature. Phi - a Coincidence?Īncient and modern architecture reflect the 'golden ratio' (1.618 length to width) and this number is remarkably close to phi (.618.) seen in nature for leaf dispersions, etc. For an aside on that, continue: Two phi‘s for the price of one?Ī student in 2001 asked about this matter of two different closely related ratios: phi vs. Specifically, it is the ratio between two parts a and b of a segment such that it is also the ratio between the whole and one of the parts: $$\frac\). The golden ratio, \(\phi\), which goes back at least to ancient Greece, has also been called the “golden mean” (because it’s a special “middle”), the “golden section” (because it is a special way of “cutting” a segment), the “divine proportion” (because it was considered perfect), and “extreme and mean ratio” (as an explicit description). I want to look at some geometrical connections and other interesting facts about this number before we get back to the Fibonacci numbers themselves and some inductive proofs involving them. We’re looking at the Fibonacci sequence, and have seen connections to a number called phi (φ or \(\phi\)), commonly called the Golden Ratio.
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