How to determine the angle within hexagonal spiral?

[Herbert Seidenberg]

Sandy:
Actually I just cleaned up the idea you had after you
came back from walking your dog.
I plotted the numbers you generated in column N and
found the trendline: 3x^2 - 3x
Your numbers in column M can be reduced to 60/{1,2,3....10}
Like you, I limited the array to 10, so that is why
my formulas work only up to 330.
Instead of LOOKUP, I used the maximum of the numbers
in column L that are less or equal to A1.
So Lukb , Luka gives the same result as your 2 LOOKUPs.

Standing on the shoulders of Giants...

 

[Eric]

Thank everyone very much for any suggestions
Eric

Hi. Just something quick-n-dirty if I understand the question:
This may be wrong.

http://www.research.att.com/~njas/sequences/A003215

Table[3*(n + 1)*n + 1, {n, 0, 9}]

{1, 7, 19, 37, 61, 91, 127, 169, 217, 271}

We note that the number of points added at each 360 rotation just increases
by 6.

Differences[%]

{6, 12, 18, 24, 30, 36, 42, 48, 54}

If given a total t (Your A1 value), then solve for n:

n -> (Sqrt(12*t - 3) - 3) / 6

So, when n=19, we've gone around 2 times:

n=19

?(Sqr(12*n - 3) - 3) / 6
2

For your example:
n=10
?(Sqr(12*n - 3) - 3) / 6
1.30277563773199

We've gone around once(6) and go four more steps during our second rotation
(Use MOD) :
Each step in degrees is:

r=2
?360/(6*r)
30

Hence 4*30 = 120

(9 is 3*30 = 90)

So, if you are looking at point 100:

n=100
?(Sqr(12*n - 3) - 3) / 6
5.2662812973354

We've gone around 5.2 times:
The firth rotation was point 91:

n=5
?3*(n + 1)*n + 1
91

Each degree difference during our 6th rotation is 10:

?360 / (6*6)
10

Angel is:

?10*(100-91+1)
100 Degrees

Again, I hope I did this correctly..:>~
--
HTH :>)
Dana DeLouis
Windows XP & Excel 2007

The formula for some angle
For the number 1 series [1,8,21,40,65 ...], the formula is N*(3N-2) on 60
Deg
For the number 2 series [2,10,24,44,70 ...], the formula is N*(3N-1) on
120
Deg
For the number 3 series [3,12,27,48,75 ...], the formula is N*(3N) on 180
Deg
For the number 4 series [4,14,30,52,80 ...], the formula is N*(3N+1) on
240
Deg
For the number 5 series [5,16,33,56,85 ...], the formula is N*(3N+2) on
300
Deg
For the number 6 series [6,18,36,60,90 ...], the formula is N*(3N+3) on
360
Deg

......16..15..14
....17..5...4...13
..18..6...0...3...12
19..7...1...2...11..26
..20..8...9...10..25
....21..22..23..24

If a number is given in cell A1, I would like to determine the angle based
on this structure of hexagonal spiral, such as 10 is the given number in
cell
A1, then 120 degree will be returned in cell B1, 9 is the given number in
cell A1, then 90 degree will be returned in cell B1.
Does anyone have any suggestions on how to determine the angle?
Thanks in advance for any suggestions
Eric

I need to

It seems like the OP did tell us but as it is gone midnight here, this
old
man is off to bed. I'll leave it to you clever folk to work it out.

--
Regards,

Sandy
In Perth, the ancient capital of Scotland
and the crowning place of kings

(e-mail address removed)
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I would think that 22 and 23 are at 80 & 100 degrees respectively. If
that
is right then the numbers on the 0, 6 18 line (reading from right to
left),
would be:

0, 6, 18, 36, 90 126, 168, 216 ie 6 * { 0, 1, 3, 6, 10, 15, 21, 28,
36}
with the interval between the numbers in braces increasing by 1 each
time.

The angle for numbers between 18 and 36 then would be 360/(36-18) = 20
Degrees.

Of course only the OP will be able to tell us.

--
HTH

Sandy
In Perth, the ancient capital of Scotland
and the crowning place of kings

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message I understood the spiral path being traced out, and I guess I can see
that
15 is at 90 degrees like 9 is... but there is (at least to my mind)
still
a problem with 22 and 23... they do not lie on a diagonal from 0
unless,
in the first 4 tiers of the spiral, they are the only number on that
diagonal. Anyway, I would like to see the OP give us a little bit more
information on how the numbers are laid down on the spiral path.

Rick


Good observation Ken. I think that you have cracked it, at least
partially, but it does not quite equate to what the OP said:

......16..15..14
....17..5...4...13
..18..6...0...3...12
19..7...1...2...11..26
..20..8...9...10..25
....21..22..23..24

1 will be inserted in 60 deg,
2 will be inserted in 120 deg,

on this structure of hexagonal spiral, such as 10 is the given
number in
cell
A1, then 120 degree will be returned in cell B1
So presumably 0, 2, 10, 24 are all on the 120 deg line

If so then surely 0,1, 8, 21 are on the 60 deg line

But if the above is true then 9 would be halfway between 60& 120 ie
90
Deg but the OP says it is equal to 80 Deg.

A1, then 120 degree will be returned in cell B1, 9 is the given
number in
cell A1, then 80 degree will be returned in cell B1.

--
HTH

Sandy
In Perth, the ancient capital of Scotland
and the crowning place of kings

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On Dec 30, 5:45 am, "Rick Rothstein \(MVP - VB\)"
I'm in agreement with you Sandy. In particular, I can't see how
number
like
15, 22 and 23 fit into the hexagonal scheme of things (they seem to
be
on
some angle other than one of the 60 degree lines); hence, I can't
figure out
how to extend the sequence of numbers in order to develop a formula
for it.

Rick



You may get an answer if you restate you request. Speaking
personally I
do not understand exactly what it is that you are asking.

--
HTH

Sandy
In Perth, the ancient capital of Scotland
and the crowning place of kings

(e-mail address removed)
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Creating a hexagonal spiral around 0,
1 will be inserted in 60 deg,
2 will be inserted in 120 deg,
3 will be inserted in 180 deg,
4 will be inserted in 240 deg,
5 will be inserted in 300 deg,
6 will be inserted in 360 deg,
and continue on the second levels as show below

......16..15..14
....17..5...4...13
..18..6...0...3...12
19..7...1...2...11..26
..20..8...9...10..25
....21..22..23..24

If a number is given in cell A1, I would like to determine the
angle
based
on this structure of hexagonal spiral, such as 10 is the given
number in
cell
A1, then 120 degree will be returned in cell B1, 9 is the given
number in
cell A1, then 80 degree will be returned in cell B1.
Does anyone have any suggestions on how to determine the angle?
Thanks in advance for any suggestions
Eric

I notice that tracing through that array of numbers from 0 to 26
results in a spiral path. But that's all I can see.

Ken Johnson

 

[Sandy Mann]

Thank you Herbert, I think that what you did was very clever.

--
Regards,

Sandy
In Perth, the ancient capital of Scotland
and the crowning place of kings

(e-mail address removed)
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