Math AOK Web Page
LJA TOK
Vocab Word: Euclidean Geometry is planar geometry (flat surface/2 dimensional), three dimensional geometry, and two column proofs. Euclidean geometry covers all geometry before the 19th century, with the exception of the parallel postulate, until Einstein and his complex theorems and mathematical exploration of gravity, light, and space.

According to three different views about the nature of mathematical truths they are either: empirical, true by definition, rational insights to universal truths
Empirical: Empirical evidence or empirical data or empirical knowledge is gained through experimentation or observation. The observation/experimentation is used to either support or reject an empirical claim. This is the opposite of rational views and claims. Rational views use only reason and reflection to support/deny claims.
True by definition: In mathematics, there are things that are always true. For example, the Pythagorean Theorem states that in all right triangles〖 a〗^2+b^2=c^2 , where a and b are the shorter sides (legs) and c is the longer side (hypotenuse). This is a mathematical definition and can be used for any triangle with one right angle.
Rational insights to universal truths: I interpret rational insights to universal truths as being additions to theorems. For example, simple trigonometry (sine, cosine, and tangent) are always used in right triangles to find un-known side lengths and angle measures.
BIG WIG
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​Leonardo Fibonacci​​
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Leonardo Fibonacci is the mathematician who is most well known for spreading the Hindu-Arabic numeral system in Europe through his book Liber Abaci, and for the number sequence named the Fibonacci sequence. In the Liber Abaci, there was an explanation of numerical numbers 0-9, place holders, fractions, and conversions of weights and measurements. All of these mathematical things became very useful in bookkeeping and calculation of interest/moneymaking.
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LINKING QUESTION
Is the Book of Nature written in the Language of Mathematics?
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I do not think the book of nature is written in the language of math. I think that mathematicians eventually find patterns just because they are looking so hard for them. That doesn’t mean that there is a mathematical explanation for every part of nature. However, it is reasonable to say that mathematics fits nature in some cases. In the case of the Fibonacci sequence, who knows if it’s coincidence that so many things in nature match this sequence, or if nature develops in a mathematical fashion.
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GUIDING QUESTION
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Is it reasonable to say mathematics Explains the Physical World?
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I don’t think it is justifiable to say that mathematics explains the entire physical world. However it does help to explain and show little bits and pieces to us that make more sense (well, maybe not). There are extreme levels of mathematics involved with physics, things like explaining the big bang with mathematical theory, black holes, etc. These mathematicians are people like Einstein and Steven Hawking. The part of mathematics in the natural world I am dealing with is a bit simpler and called the Fibonacci sequence. This simple sequence of adding a number to the one before it in the sequence starting with one and two (1, 2, 3, 5, 8…) has shown up in all parts of nature, from seashells to the number of pedals on a flower. The Fibonacci sequence doesn’t necessarily explain mathematics in the physical world, but it rather connects the two worlds whether it is coincidence or not.

Nick Whitcomb's AoK Mathematics Website
PARADIGM SHIFT
​Where it’s been: The Fibonacci sequence was known by Indian mathematicians as early as the 6th century, but Leonardo Fibonacci introduced this sequence and its mystery to the western world in the 1200s with his book Liber Abaci.
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Where it is now: Currently the Fibonacci numbers and golden ratio is found EVERYWHERE. It is mysterious, and no one has really put any answers to why it is so.
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Where it is going: I think the future link of the golden ratio and the Fibonacci sequence to nature will be explained in some complex mathematician way. I am curious to know what makes this sequence so highly repeated throughout nature.
DIFFERENT PERSPECTIVES
There are a few different perspectives people could have on the Fibonacci sequence. The first is that mathematicians are defining nature (the patterns could be coincidence and mathematicians are finding patterns out of nothing). Or nature is defining mathematicians (nature does grow in a mathematical pattern and this is how mathematicians know what they know). Or that we don’t know whether it’s math defining nature or nature defining math, but think it is interesting either way.
ABOUT ME

My name is Nick Whitcomb and I like to paddle, kayak, and surf. I am a senior this year at Le Jardin and my favorite classes are biology math and chemistry. I enjoy mathematics because there is almost always a correct answer that can be achieved by following steps to dissect it. For this reason, I wrote my extended essay on a topic in mathematics.