The Geometry of the Times Square Ball

After watching the traditional New Year's Day ball drop in New York City (on television; I wouldn't stand in the rain for hours even if I were capable of doing so), I became curious about the construction of the ball, and did some research on the Internet, finding https://www.azocleantech.com/article.aspx?ArticleID=311 and other descriptions.

 

[Images derived from Times_Square_Ball_2010.jpg by Susan Serra, CKD from Long Island, USA - 222 (2), CC BY-SA 2.0, https://commons.wikimedia.org/w/index.php?curid=23313305]

It is described as having "a diameter of 12 feet and weighing 11,575 pounds" and is "covered with 2,688 Waterford Crystals", also called "triangular crystal panels".  It has a total of "32,256 Philips Luxeon Rebel energy-saving LEDs", and from this I figured that there are 12 LEDs per triangular crystal panel.

There are 672 "LED units", each having 48 LED lights, 12 each of "red, blue, green and white colors".  672 times 48 is 32256, confirming the total LEDs.  We can also conclude that each "LED unit" has 4 triangular crystal panels, because there are 12 LEDs per crystal panel.

In the photo of the Times Square Ball, we can see triangles framed by the black aluminum struts; we will call these 'frame' triangles.  Each frame triangle is divided into four 'color' triangles which apparently have independently selectable colors.  And if you look carefully, each color triangle is divided into four 'tiny' triangles.  So which of these triangle sizes is a "crystal panel"?

Geodesic spheres are nearly always based on the geometry of an icosahedron, generally by dividing its triangular faces into smaller triangles.  There are 20 triangles in a icosahedron; 12 vertices, and 30 edges, and 5 edges meet at each vertex.  A typical geodesic sphere is full of triangles, and in most places, a cluster of six triangles around a vertex form a hexagon.  But in rare places, a cluster of five triangles around a vertex will form a pentagon.

Finding these pentagons is the trick to discovering the icosahedron from which the geodesic sphere was formed, because the centers of these pentagons are the vertices of the icosahedron.  You can impress your friends by 'counting' the number of triangles in the complete 'sphere' that you see (even though it is not actually complete or not all visible) in two or three seconds, as follows:  Find two pentagons and along the line that connects their centers, count the number of edges of the small triangles.  Square this number and multiply by 20.  That's it.  If you counted 10 edges, the answer is 10x10x20=2000.  In a few seconds, you have 'counted' 2000 triangles!

In the photo at right, we have located the centers of three pentagons, and thus one triangle of the icosahedron.  Three struts make one side of this triangle of the icosahedron, making 3x3=9 frame triangles in  one triangle of the icosahedron, or 9x20=180 frame triangles in the complete geodesic sphere, not enough for "672 LED units".  But there are four color triangles in each frame triangle, so there are 4x180=720 color triangles in the complete geodesic sphere, 48 more than the "672 LED units".  But there must be two holes in the sphere to allow it to slide down the supporting pole, so we can guess that each hole must account for 24 color triangles, or 24  "LED units", or 24/4=6 frame triangles.  A hexagon of 6 frame triangles is marked on the next photo below, which looks plausibly the right size for a hole.  A study of the symmetries of the icosahedron confirms that such a hexagon is matched by one on the opposite side of the sphere.

So we conclude that each color triangle corresponds to an LED unit, which has 48 LED lights.  And each tiny triangle must correspond to a Waterford crystal panel, which  has 12 LED lights, 3 LEDs of each color, which can be neatly arranged in a triangle.  I have seen a number of products using LEDs which embed LEDs in a plastic sheet to distribute the light over the area; so I guess that LEDs are similarly embedded in the Waterford Crystals.

We have been ignoring one small but interesting detail.  When a flat triangle of an icosahedron is divided into smaller triangles (144 Waterford Crystals in this case) by dividing the edge length by an integer value (12 in this case), the big triangle remains flat.  What is actually done is that the smaller triangles are made a tiny bit larger; making the big triangle bulge out just enough so that all the triangle corners (vertices) lie perfectly on a spherical surface; which is how geodesic spheres look almost like real spheres.  The calculation of this adjustment involves trigonometry; need I say more?

Someone gave me the following link to an article listing "Seven mind-scrambling science ideas only geniuses can understand". 
http://metro.co.uk/2014/12/13/seven-mind-scrambling-science-ideas-only-geniuses-can-understand-4985026/

The article said that "The world’s scientists don’t just sit around doing really hard maths for fun – they also believe some truly crazy ideas."

My comments on these seven ideas follow. but first, keep in mind that science does not explain everything; we must also deal with metaphysics, philosophy, and theology here.  Science has been defined as the study of all that is material (mass-energy-space-time), so only materialists, those that ascribe to the metaphysical notion that everything is material (non-material things are imaginary) -- only materialists believe that science can explain everything.

In this modern era, we also study information just as intently, but information is not made of mass-energy-space-time -- mass and energy are only used in arbitrarily different ways for tranport of information through either space (communication) or time (storage).  Information is observed only where life is observed.  This frustrates the materialists, who resort to all kinds of stupidity trying to explain life.  (For more on this, read my blogs such as http://jamesmclark.blogspot.com/2005/07/in-beginning-was-information.html,
http://jamesmclark.blogspot.com/2007/06/is-encoded-information-essential-part.html, http://jamesmclark.blogspot.com/2008/09/life-is-more-than-chemistry.html, and.http://jamesmclark.blogspot.com/2010/03/digital-control-of-life.html.)  To define information theory as part of science, you either have to change the definition of 'material' or stop being a materialist.

1. Time goes slower on the Space Station

"This isn’t just theory – it’s actually measurable."

A proven fact.  From working on GPS, I know that when we make and test the time-keeping system of a GPS satellite on the ground, we have to 'set the clock' a little faster on the ground so that after it is launched, it will agree with our global time-keeping system on the ground.  The 'correction' is mostly to account for the speed of the satellite (a 12-hour orbit), but also to account for the decreased gravity (altitude).

2. We are almost certainly living in the Matrix

"British philosopher Nick Bostrom claims that we are probably living inside a Matrix-style simulation."

A conjecture.  I think that if Nick Bostrom studied complexity theory, he would change his mind.

3. In a class of 25 children, two will probably share a birthday

"But it’s actually more likely than not that two will share a birthday – a chance of 57%."

Yes, the probabilty is closest to 50% for a class of 23; for 25, the probability goes up a little.  As I recall, it is a difficult calculation requiring surprisingly high precision. 

4. There is more than one universe

"There are billions, according to a theory which is accepted by ‘nearly all’ cosmologists."

A conjecture.  This is based on the accepted fact that the universe that we observe is finite.  Most people would think that this requires that the universe has an edge, or boundary.  But topologists know that a 3D space can be folded on itself edgelessly, like the 2D surface of a sphere, for example.  So, some far-away galaxy might be our local galaxy (the Milky Way) many, many years ago.  This conjecture is scientifically unprovable, but many Christians assume that heaven is in another universe.  (We really don't know.)

5. There are more than four dimensions

"...there are either dimensions too small for us to see, or that our 4D world exists on a ‘brane’ which floats in another, higher-dimensional world."

Only in subatomic physics.  The extra dimensions 'fold up' within the tiny spaces of atoms.

6. No one knows what a computer is

"Computing professors worry about this stuff. Is an abacus a computer? Is a sundial?"

This is really just saying that there isn't yet a universally agreed definition of a computer, especially of very small ones.  I once designed a computer with only one bit of internal memory and unlimited external memory, which I started to build but never finished.  It would have been practically useless, taking a very long time to compute something very small.

7. The universe should not exist at all

"The universe may not have started with a Big Bang", the article says.

"Prof Mersini-Houghton’s calculations seem to prove that when a dying star collapses in on itself, it does not shrink down to become a ‘singularity’, ... what we know as a black hole.

Instead, the radiation that escapes from dying stars robs them of their mass, so they never have enough gravity to form black holes.

This creates a major problem. The ‘fuse’ that lit the Big Bang is supposed to have been a singularity – something which has now been proven not to exist."

I think this conundrum is a consequence of relying only on science.  In other words, it results from an inherent limitation of science.  It demonstrates a fundamental question that science alone cannot explain.  All you can conclude is that something outside of science created the universe.  If you add information theory to your thinking toolbox, you can conclude that 'that something' had an enormous amount of information, since we know that the material universe can destroy information but can never create it.  And since information is observed only where life is observed, we can also conclude that 'that something' is living.  It takes theology to go further than that, and the Bible to go in the right direction.

How to use the "WHAT IS NUMBER" T-shirt

I have created T-shirt designs with a mathematical theme, and I sell them on http://www.zazzle.com/mathart.  Lately, I created a magic trick using mathematics and some other tricks for 'guessing' a number that a volunteer has secretly chosen.  Then I adapted this magic trick, in a simpler form, for a T-shirt design.  For those who buy the "WHAT IS NUMBER" T-shirt (or receive one as a gift), here is an explanation of how to use the T-shirt as a magic prop.  To buy the T-shirt, go to http://www.zazzle.com/what_number_magic_shirt-235678841307212061?rf=238851269352169293 (you can customize the shirt.)

You ask a volunteer to choose a number in the range 1 to 144, but to keep it secret. You ask him to find his number on the front of the shirt, and to tell you the letter in the same box as his number. For example, he tells you "U". You ask him to also find his number on the back of the shirt, and to tell you the corresponding letter. While he is finding his number on the back, you translate the "U" to 4, using the mnemonic "foUr", and multiply the 4 by 12, getting 48. He reports "S" for the back, and you translate the "S" to 6, using the mnemonic "Six", and add the 6 to the 48, getting 54. You tell him his number is 54, without (of course) telling him how you derived that number, and he is amazed.

Optionally, you may ask the volunteer to write down his number on a card and to show the card to others nearby.  This allows others to participate by verifying that your 'guess' is correct.  It is obvious that you cannot see the back of the T-shirt, but you should not look down at the front of the T-shirt, either.  Look straight at the volunteer, or the audience, or look up, or close your eyes, or be blindfolded.  The audience may think you have the entire shirt memorized, but that is not necessary.

What is the complete method, and how does it work? The method is based on the following matrix and the cipher that follows.

Matrix of numbers 1 to 144 with rows and columns labelled

	B	I	W	H	U	R	S	E	A	N	T	M
B	144	1	2	3	4	5	6	7	8	9	10	11
I	12	13	14	15	16	17	18	19	20	21	22	23
W	24	25	26	27	28	29	30	31	32	33	34	35
H	36	37	38	39	40	41	42	43	44	45	46	47
U	48	49	50	51	52	53	54	55	56	57	58	59
R	60	61	62	63	64	65	66	67	68	69	70	71
S	72	73	74	75	76	77	78	79	80	81	82	83
E	84	85	86	87	88	89	90	91	92	93	94	95
A	96	97	98	99	100	101	102	103	104	105	106	107
N	108	109	110	111	112	113	114	115	116	117	118	119
T	120	121	122	123	124	125	126	127	128	129	130	131
M	132	133	134	135	136	137	138	139	140	141	142	143

The front of the T-shirt indicates which row, and the back of the T-shirt indicates which column. The letters are translated to numbers according to the following cipher. The mnemonics are an aid to memorizing the cipher.

Cipher and mnemonics:

W	2	tWo, first spelled number with W
H	3	tHree, first spelled number with H
A	8	sounds like Ate
T	10	Ten, fourth spelled number with T
I	1	I looks like 1
S	6	Six, first spelled number with S
N	9	NiNe, first spelled number with two N's
U	4	foUr, first spelled number with U
M	11	eleven – Melvin (similar sound)
B	0	B00, surprise, O looks like 0
E	7	sEvEn, first spelled number with two separate E's
R	5	fiveR, slang for 5-dollar bill

After translating the front and back letters to numbers Fr and Bk, calculate this in your head:

Result = 12 * Fr + Bk

You can calculate the 12 * Fr part while the back number is being found.

EXCEPTION: If the Result is zero, change it to 144. 

The Reverse Trick

You can do the magic trick in reverse.  That is, given 54, for example, you can determine where it is found without looking.  You divide 54 by 12, getting the quotient 4 and remainder 6.  Using the cipher, you convert the quotient 4 to U, the position on the front of the shirt, and you convert the remainder 6 to S, the position on the back of the shirt.

Another Trick

For any letter, there is a number that appears under that letter on both the front and the back of the T-shirt.  So you can ask a volunteer to choose a letter, and then you choose a number that appears under that letter on both the front and the back.  To do this, it will help to be familiar with the multiples of 13.  (Most people are not, and as a result, think that 91 is a prime number.)

Given a letter, use the cipher to convert the letter to a number, then multiply the number by 13.  For example, suppose the letter is E.  According to the cipher, the corresponding number is 7.  7 times 13 is 91, so you declare that 91 is found in box E on both the front and the back of the T-shirt.

The equivalence of 0 and 144 applies to all tricks.