How to determine the angle within hexagonal spiral?

OK so I had to take the dog for a walk first and got to thinking about this:

In K2 enter 1 and K3 enter 3. In K4 enter the formula:

=(K3-K2+1)+K3

and copy down as far as needed,

In L2 enter 0 and in L3 the formula:

=K2*6

and copy down as far as in Column K. These are the numbers along the 0/6
line

In M2 enter the formula:

=360/(L3-L2)
and copy down to one row short of the the othe rtwo columns.

With the required number in A1 enter in B1:

=(A1-INDEX(L2:L10,MATCH(LOOKUP(A1,L2:L10),L2:L10)))*LOO KUP(A1,L2:L9,M2:M9)

This should be the degrees that you are looking for.

There may of course be more elegant ways of doing it.

--
HTH

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


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"Sandy Mann" wrote in message
...
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


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"Sandy Mann" wrote in message
...
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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"Rick Rothstein (MVP - VB)" wrote in
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


"Sandy Mann" wrote in message
...
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


Replace @mailinator.com with @tiscali.co.uk


"Ken Johnson" wrote in message
...
On Dec 30, 5:45 am, "Rick Rothstein \(MVP - VB\)"
wrote:
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

"Sandy Mann" wrote in message

...

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


Replace @mailinator.com with @tiscali.co.uk

"Eric" wrote in message
...
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

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

"Sandy Mann" wrote:

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


Replace @mailinator.com with @tiscali.co.uk


"Sandy Mann" wrote in message
...
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


Replace @mailinator.com with @tiscali.co.uk


"Rick Rothstein (MVP - VB)" wrote in
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


"Sandy Mann" wrote in message
...
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


Replace @mailinator.com with @tiscali.co.uk


"Ken Johnson" wrote in message
...
On Dec 30, 5:45 am, "Rick Rothstein \(MVP - VB\)"
wrote:
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

"Sandy Mann" wrote in message

...

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


Replace @mailinator.com with @tiscali.co.uk

"Eric" wrote in message
...
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

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


"Eric" wrote in message
...
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

"Sandy Mann" wrote:

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


Replace @mailinator.com with @tiscali.co.uk


"Sandy Mann" wrote in message
...
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


Replace @mailinator.com with @tiscali.co.uk


"Rick Rothstein (MVP - VB)" wrote in
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


"Sandy Mann" wrote in message
...
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


Replace @mailinator.com with @tiscali.co.uk


"Ken Johnson" wrote in message
...
On Dec 30, 5:45 am, "Rick Rothstein \(MVP - VB\)"
wrote:
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

"Sandy Mann" wrote in message

...

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


Replace @mailinator.com with @tiscali.co.uk

"Eric" wrote in message
...
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

So if I follow you correctly, changing it into one formula gives us:

=360/(6*(INT((SQRT(12*A1 - 3)-3) / 6)+1))*((A1-(3*(INT((SQRT(12*A1 - 3) - 3)
/ 6) + 1)*INT((SQRT(12*A1 - 3) - 3) / 6) + 1))+1)

I'll leave it to Rick to cut out any extra key strokes g

--
Regards,

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


Replace @mailinator.com with @tiscali.co.uk


"Dana DeLouis" wrote in message
...
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


"Eric" wrote in message
...
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

"Sandy Mann" wrote:

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


Replace @mailinator.com with @tiscali.co.uk


"Sandy Mann" wrote in message
...
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


Replace @mailinator.com with @tiscali.co.uk


"Rick Rothstein (MVP - VB)" wrote
in
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


"Sandy Mann" wrote in message
...
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


Replace @mailinator.com with @tiscali.co.uk


"Ken Johnson" wrote in message
...
On Dec 30, 5:45 am, "Rick Rothstein \(MVP - VB\)"
wrote:
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

"Sandy Mann" wrote in message

...

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


Replace @mailinator.com with @tiscali.co.uk

"Eric" wrote in message
...
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

Hi Sandy. I think that looks good. Very nice.

That link had another link to the following:

http://www.research.att.com/~njas/sequences/a3215.gif

The op's diagram started with 0, and the zero angle begins in the -x
direction.
The picture above begins with 1, and the
{1, 7, 19, 37, 61,...} sequence is at a different angle.
The op's just wants to rotate that diagram to have the zero angle on the -x
direction.
If we add 1 to each point in the op's diagram, and rotate, then the two
diagrams will match.

--
Dana DeLouis



"Sandy Mann" wrote in message
...
So if I follow you correctly, changing it into one formula gives us:

=360/(6*(INT((SQRT(12*A1 - 3)-3) / 6)+1))*((A1-(3*(INT((SQRT(12*A1 - 3) -
3) / 6) + 1)*INT((SQRT(12*A1 - 3) - 3) / 6) + 1))+1)

I'll leave it to Rick to cut out any extra key strokes g

--
Regards,

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


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"Dana DeLouis" wrote in message
...
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


"Eric" wrote in message
...
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

"Sandy Mann" wrote:

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


Replace @mailinator.com with @tiscali.co.uk


"Sandy Mann" wrote in message
...
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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"Rick Rothstein (MVP - VB)" wrote
in
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


"Sandy Mann" wrote in message
...
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


Replace @mailinator.com with @tiscali.co.uk


"Ken Johnson" wrote in message
...
On Dec 30, 5:45 am, "Rick Rothstein \(MVP - VB\)"
wrote:
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

"Sandy Mann" wrote in message

...

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


Replace @mailinator.com with @tiscali.co.uk

"Eric" wrote in message
...
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

Yet another solution with defined names:
Array ={1;2;3;4;5;6;7;8;9;10}
Square =3*Array*(Array-1)
Luka =MAX(Square*(Square=$A$1))
Lukb =60/MAX(Array*(Square=$A$1))
B1 =($A$1-Luka)*Lukb

Good job Sandy! The only simplification I can make (besides removing all
those spaces) is to divide out the 6 from the denominator. In addition, I
have a personal preference for ordering chained multiplications and
divisions to put the multiplication first; so, for the layout of A/B*C that
you have, I prefer to rearrange that to A*C/B for clarity. Hence, I would
present your formula (with the aforementioned division carried out) like
this...

=60*((A1-(3*(INT((SQRT(12*A1-3)-3)/6)+1)*INT((SQRT(12*A1-3)-3)/6)+1))+1)/(INT((SQRT(12*A1-3)-3)/6)+1)

Rick


"Sandy Mann" wrote in message
...
So if I follow you correctly, changing it into one formula gives us:

=360/(6*(INT((SQRT(12*A1 - 3)-3) / 6)+1))*((A1-(3*(INT((SQRT(12*A1 - 3) -
3) / 6) + 1)*INT((SQRT(12*A1 - 3) - 3) / 6) + 1))+1)

I'll leave it to Rick to cut out any extra key strokes g

--
Regards,

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


Replace @mailinator.com with @tiscali.co.uk


"Dana DeLouis" wrote in message
...
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


"Eric" wrote in message
...
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

"Sandy Mann" wrote:

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


Replace @mailinator.com with @tiscali.co.uk


"Sandy Mann" wrote in message
...
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


Replace @mailinator.com with @tiscali.co.uk


"Rick Rothstein (MVP - VB)" wrote
in
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


"Sandy Mann" wrote in message
...
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


Replace @mailinator.com with @tiscali.co.uk


"Ken Johnson" wrote in message
...
On Dec 30, 5:45 am, "Rick Rothstein \(MVP - VB\)"
wrote:
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

"Sandy Mann" wrote in message

...

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


Replace @mailinator.com with @tiscali.co.uk

"Eric" wrote in message
...
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

Here's a vba function if you want to go that route:

Function Angle(x) As Double
Dim n As Double
n = Int((Sqr(12 * x - 3) - 3) / 6)
Angle = 60 * (x / (n + 1) - 3 * n)
End Function

--
Dana DeLouis


"Rick Rothstein (MVP - VB)" wrote in
message ...
Good job Sandy! The only simplification I can make (besides removing all
those spaces) is to divide out the 6 from the denominator. In addition, I
have a personal preference for ordering chained multiplications and
divisions to put the multiplication first; so, for the layout of A/B*C
that you have, I prefer to rearrange that to A*C/B for clarity. Hence, I
would present your formula (with the aforementioned division carried out)
like this...

=60*((A1-(3*(INT((SQRT(12*A1-3)-3)/6)+1)*INT((SQRT(12*A1-3)-3)/6)+1))+1)/(INT((SQRT(12*A1-3)-3)/6)+1)

Rick


"Sandy Mann" wrote in message
...
So if I follow you correctly, changing it into one formula gives us:

=360/(6*(INT((SQRT(12*A1 - 3)-3) / 6)+1))*((A1-(3*(INT((SQRT(12*A1 - 3) -
3) / 6) + 1)*INT((SQRT(12*A1 - 3) - 3) / 6) + 1))+1)

I'll leave it to Rick to cut out any extra key strokes g

--
Regards,

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


Replace @mailinator.com with @tiscali.co.uk


"Dana DeLouis" wrote in message
...
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


"Eric" wrote in message
...
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

"Sandy Mann" wrote:

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


Replace @mailinator.com with @tiscali.co.uk


"Sandy Mann" wrote in message
...
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


Replace @mailinator.com with @tiscali.co.uk


"Rick Rothstein (MVP - VB)" wrote
in
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


"Sandy Mann" wrote in message
...
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


Replace @mailinator.com with @tiscali.co.uk


"Ken Johnson" wrote in message
...
On Dec 30, 5:45 am, "Rick Rothstein \(MVP - VB\)"
wrote:
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

"Sandy Mann" wrote in message

...

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


Replace @mailinator.com with @tiscali.co.uk

"Eric" wrote in message
...
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

"Rick Rothstein (MVP - VB)" wrote in
message ...
Good job Sandy!

All I did was to blindly transpose Dana's method into a formula. I did not
understand it all sufficiently well to start messing about with it - so I
passed it on to you g

Actually Herbert's Defined Name formula, which is better and which I equally
well do not understand, (I would be grateful for an explanation from Herbert
or anyone else), but seems to return wrong results for points above point
330. I say this because all results after point 330 are 6 Deg increases.

--
Regards,

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


Replace @mailinator.com with @tiscali.co.uk


"Rick Rothstein (MVP - VB)" wrote in
message ...
Good job Sandy! The only simplification I can make (besides removing all
those spaces) is to divide out the 6 from the denominator. In addition, I
have a personal preference for ordering chained multiplications and
divisions to put the multiplication first; so, for the layout of A/B*C
that you have, I prefer to rearrange that to A*C/B for clarity. Hence, I
would present your formula (with the aforementioned division carried out)
like this...

=60*((A1-(3*(INT((SQRT(12*A1-3)-3)/6)+1)*INT((SQRT(12*A1-3)-3)/6)+1))+1)/(INT((SQRT(12*A1-3)-3)/6)+1)

Rick


"Sandy Mann" wrote in message
...
So if I follow you correctly, changing it into one formula gives us:

=360/(6*(INT((SQRT(12*A1 - 3)-3) / 6)+1))*((A1-(3*(INT((SQRT(12*A1 - 3) -
3) / 6) + 1)*INT((SQRT(12*A1 - 3) - 3) / 6) + 1))+1)

I'll leave it to Rick to cut out any extra key strokes g

--
Regards,

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


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"Dana DeLouis" wrote in message
...
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


"Eric" wrote in message
...
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

"Sandy Mann" wrote:

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


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"Sandy Mann" wrote in message
...
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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"Rick Rothstein (MVP - VB)" wrote
in
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


"Sandy Mann" wrote in message
...
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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"Ken Johnson" wrote in message
...
On Dec 30, 5:45 am, "Rick Rothstein \(MVP - VB\)"
wrote:
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

"Sandy Mann" wrote in message

...

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


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"Eric" wrote in message
...
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

I like it! I don't understand it but I like it!

--

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


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"Dana DeLouis" wrote in message
...
Here's a vba function if you want to go that route:

Function Angle(x) As Double
Dim n As Double
n = Int((Sqr(12 * x - 3) - 3) / 6)
Angle = 60 * (x / (n + 1) - 3 * n)
End Function

--
Dana DeLouis


"Rick Rothstein (MVP - VB)" wrote in
message ...
Good job Sandy! The only simplification I can make (besides removing all
those spaces) is to divide out the 6 from the denominator. In addition, I
have a personal preference for ordering chained multiplications and
divisions to put the multiplication first; so, for the layout of A/B*C
that you have, I prefer to rearrange that to A*C/B for clarity. Hence, I
would present your formula (with the aforementioned division carried out)
like this...

=60*((A1-(3*(INT((SQRT(12*A1-3)-3)/6)+1)*INT((SQRT(12*A1-3)-3)/6)+1))+1)/(INT((SQRT(12*A1-3)-3)/6)+1)

Rick


"Sandy Mann" wrote in message
...
So if I follow you correctly, changing it into one formula gives us:

=360/(6*(INT((SQRT(12*A1 - 3)-3) / 6)+1))*((A1-(3*(INT((SQRT(12*A1 -
3) - 3) / 6) + 1)*INT((SQRT(12*A1 - 3) - 3) / 6) + 1))+1)

I'll leave it to Rick to cut out any extra key strokes g

--
Regards,

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


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"Dana DeLouis" wrote in message
...
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


"Eric" wrote in message
...
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

"Sandy Mann" wrote:

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


Replace @mailinator.com with @tiscali.co.uk


"Sandy Mann" wrote in message
...
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


Replace @mailinator.com with @tiscali.co.uk


"Rick Rothstein (MVP - VB)"
wrote in
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


"Sandy Mann" wrote in message
...
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


Replace @mailinator.com with @tiscali.co.uk


"Ken Johnson" wrote in message
...
On Dec 30, 5:45 am, "Rick Rothstein \(MVP - VB\)"
wrote:
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

"Sandy Mann" wrote in message

...

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


Replace @mailinator.com with @tiscali.co.uk

"Eric" wrote in message
...
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