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Akano's Blog



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That moment when you realize

Posted by Akano , in Life Feb 17 2017 · 77 views
Coincidence?!, Yes and 1 more...
...that your two blog entries on pentagrams are six days short of being exactly one year apart from one another.

lel

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Equation of the Day #19: Golden Pentagrams

Posted by Akano , in Math/Physics Feb 10 2017 · 123 views
Pentagram, Five, Golden Ratio and 2 more...

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Ah, the pentagram, a shape associated with a variety of different ideas, some holy, some less savory. But to me, it's a golden figure, and not just because of how I chose to render it here. The pentagram has a connection with the golden ratio, which is defined as

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This number is tied to the Fibonacci sequence and the Lucas numbers and seems to crop up a lot in nature (although how much it crops up is disputed). It turns out that the various line segments present in the pentagram are in golden ratio with one another.

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In the image above, the ratio of red:green = green:blue = blue:black is the golden ratio. The reason for this is not immediately obvious and requires a bit of digging, but the proof is fairly straightforward and boils down to a simple statement.

First, let's consider the pentagon at the center of the pentagram. What is the angle at each corner of a pentagon? There's a clever way to deduce this. It's not quite clear what the interior angle is (that is, the angle on the inside of the shape at an individual corner), but it's quite easy to get the exterior angle.


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The exterior angle of the pentagon (which is the angle of the base of the triangles that form the points of the pentagram) is equal to 1/5 of a complete revolution around the circle, or 72°. For the moment, let's call this angle 2θ. To get the angle that forms the points of the pentagram, we need to invoke the fact that the sum of all angles in a triangle must equal 180°. Thus, the angle at the top is 180° – 72° – 72° = 36°. This angle I will call θ. While I'm at it, I'm going to label the sides of the triangle x and s (the blue and black line segments from earlier, respectively).

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We're nearly there! We just have one more angle to determine, and that's the first angle I mentioned – the interior angle of the pentagon. Well, we know that the interior angle added to the exterior angle must be 180°, since the angles both lie on a straight line, so the interior angle is 180° – 72° = 108° = 3θ. Combining the pentagon and the triangle, we obtain the following picture.

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Now you can probably tell why I labeled the angles the way I did; they are all multiples of 36°. What we want to show is that the ratio x/s is the golden ratio. By invoking the Law of sines on the two isosceles triangles in the image above, we can show that

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This equation just simplifies to sin 2θ = sin 3θ. With some useful trigonometric identities, we get a quadratic equation which we can solve for cos θ.

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Solving this quadratic equation yields

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which, when taken together with the equation for x/s, shows that x/s is indeed the golden ratio! Huzzah!

The reason the pentagram and pentagon are so closely tied to the golden ratio has to do with the fact that the angles they contain are multiples of the same angle, 36°, or one-tenth of a full rotation of the circle. Additionally, since the regular dodecahedron (d12) and regular icosahedron (d20) contain pentagons, the golden ratio is abound in them as well.

As a fun bonus fact, the two isosceles triangles are known as the golden triangle (all acute angles) and the golden gnomon (obtuse triangle), and are the two unique isosceles triangles whose sides are in golden ratio with one another.

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So the next time you see the star on a Christmas tree, the rank of a military officer, or the geocentric orbit of Venus, think of the number that lurks within those five-pointed shapes.

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Shoveled Knight

Posted by Akano , in Video Games Jan 24 2017 · 103 views
Shovel Knight, The Enchantress and 3 more...

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I'm kinda in love with this game. Also the soundtrack. Virt can lay down some sick tracks, yo.

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Me playing Pokémon Moon

Posted by Akano , in Video Games, Life Dec 02 2016 · 135 views

Me: "Hau just handed me a person named Max Potion. I think that's human trafficking."

KK: *bursts into laughter*

Me: *joins in laughter because he didn't expect that joke to land*

Another typical Friday evening.

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I had Christmas down in Africa

Posted by Akano , Dec 01 2016 · 145 views
Five Golden Rings and 3 more...


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Nothing worth having comes without some kind of fight

Posted by Akano , in Life Nov 09 2016 · 131 views

You've gotta kick at the darkness 'til it bleeds daylight.



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Exercise those rights

Posted by Akano , in Life Nov 08 2016 · 172 views
Yes We Stan, Vote, Be heard
Go vote.

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Spirit of Justice

Posted by Akano , in Video Games Oct 26 2016 · 173 views
Phoenix Wright, Apollo Justice and 3 more...
Finally, we got an Ace Attorney game with fun Apollo Justice cases! What was once thought impossible has been achieved!

Seriously, though, Spirit of Justice ruled. It definitely made up for Dual Destinies' lack of awesome. Also, the puns were taken up to eleven. No complaints here.

Although Case 4 was VERY out of place. Basically, the only thing worthwhile there was Blackquill's appearance. I didn't really like how they portrayed Athena in that one. <_<

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Equation of the Day #18: 12

Posted by Akano , in Math/Physics Sep 07 2016 · 195 views
star polygon, dodecagon, music and 2 more...

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Yesterday I stumbled across this image (which I recreated and cleaned up a bit). It's a beautiful image. Arranged around the edge is the circle of fifths, which in music is a geometric representation of the twelve tones of the Western scale arranged so the next note is seven semitones up (going clockwise in this figure). The notes are all connected in six different ways to the other notes in the "circle," known as intervals, which are color-coded at the bottom. I thought, "Wow, this is a really cool way to represent this geometrically. How neat!" However, I found the original website that the image came from, and it's a pseudoscience site that talks about the fractal holographic nature of the universe. While fractals do show up in Nature a lot, and there are legitimate theories proposing that the Universe may indeed be a hologram, what their site is proposing is, to put it lightly, utter nonsense. But instead of tearing their website apart (which would be rather cathartic), I instead want to point out the cool math going on here, because that sounds more fun!

Looking at the bottom of the graphic, you'll notice six figures. The first (in red) is a regular dodecagon, a polygon with twelve equal sides and angles. This shape is what forms the circle of fifths. The rest of the shapes in the sequence are dodecagrams, or twelve-pointed stars. The first three are stars made up of simpler regular polygons; the orange star is made up of two hexagons, the yellow is made up of three squares, and the green one is made up of four triangles. The final dodecagram (in purple) can be thought of as made up of six straight-sided digons, or line segments. These shapes point to the fact that twelve is divisible by five unique factors (not including itself): one set of twelve, two sets of six, three sets of four, four sets of three, and six sets of two! You could say that the vertices of the dodecagon finalize the set as twelve sets of one, but they're not illustrated in this image. So really, this image has less to do with musical intervals and more to do with the number 12, which is a rather special number. It is a superior highly composite number, which makes it a good choice as a number base (a reason why feet are divided into twelve inches, for instance, or why our clocks have twelve hours on their faces).

The final dodecagram in cyan is not made up of any simpler regular polygons because the number 12 is not divisible by five. If you pick a note in the circle of fifths to start on, you'll notice that the two cyan lines that emanate from it connect to notes that are five places away on the "circle," hence the connection to the number 5. In fact, it would be far more appropriate to redraw this figure with a clock face.

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This new image should shed some more light on what's really going on. The dodecagrams each indicate a different map from one number to another, modulo 12. The only reason this is connected to music at all is due to the fact that a Western scale has twelve tones in it! If we used a different scale, such as a pentatonic scale (with five tones, as the name would suggest), we'd get a pentagon enclosing a pentagram. Really, this diagram can be used to connect any two elements in a set of twelve. The total number of connecting lines in this diagram, then, are

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where the notation in parentheses is "n choose 2," and Tn is a triangular number. This figure is known in math as K12, the complete graph with twelve nodes. And it's gorgeous.

So while this doesn't really have anything to do with music or some pseudoscientific argument for some fancy-sounding, but ultimately meaningless, view on the universe, it does exemplify the beauty of the number 12, and has a cool application to the circle of fifths. :)

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Just beat the original Legend of Zelda last night

Posted by Akano , in Video Games Aug 09 2016 · 210 views
Zelda, Gannon banned, Link and 2 more...
Am I a true fan now?

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Oak Log Bans

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About Me

Akano
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Premier Members
Stone Champion Nuva
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1,500+ posts
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+2 for Premier Membership
+1 from Pohuaki for reporting various things in Artwork


Name: Akano
Real Name: Forever Shrouded in Mystery :P
Age: 28
Gender: Male
Likes: Science, Math, LEGO, Bionicle, Comics, Yellow, Voice Acting, Pixel Art, Video Games
Notable Facts: One of the few Comic Veterans still around
Has been a LEGO fan since ~1996
Bionicle fan from the beginning
Twitter: @akanotoe

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My Lovely Topics

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Hieroglyphs And The Like

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Recent Comments

Approvals

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