How to determine the angle within hexagonal spiral?

[Eric]

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

 

[Sandy Mann]

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

 

[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

 

[Ken Johnson]

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

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

 

[Ken Johnson]

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

Also, maybe that 80 degrees is a typo, ie 9 is the given number in
cell A1 then 180 degrees will be returned in cell B1.

Ken Johnson

 

[Sandy Mann]

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

--
HTH

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

(e-mail address removed)
Replace @mailinator.com with @tiscali.co.uk

 

[Rick Rothstein \(MVP - VB\)]

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

 

[Eric]

Thank everyone very much for any suggestions

Yes, 9 would be halfway between 60 & 120 ie 90 Deg, not 80 Deg.
Along the 60 Deg, there are 1,8,21
Along the 120 Deg, there are 2,10,24
Since there is only 22 and 23 between 21 at 60 Deg and 24 at 120 Deg, then
The angle for 22 and 23 can be determined by dividing the angle between 60
and 120 Deg, therefore the angle for 22 will be 80 Deg and the angle for 23
will be 100 Deg.
Does anyone have any suggestions?
Thank everyone for any suggestions
Eric

 

[Sandy Mann]

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

(e-mail address removed)
Replace @mailinator.com with @tiscali.co.uk

 

[Sandy Mann]

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

 

[Sandy Mann]

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)))*LOOKUP(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

(e-mail address removed)
Replace @mailinator.com with @tiscali.co.uk

 

[Eric]

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

 

[Dana DeLouis]

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

 

[Herbert Seidenberg]

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

 

[Rick Rothstein \(MVP - VB\)]

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

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

(e-mail address removed)
Replace @mailinator.com with @tiscali.co.uk

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


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

(e-mail address removed)
Replace @mailinator.com with @tiscali.co.uk


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


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

(e-mail address removed)
Replace @mailinator.com with @tiscali.co.uk


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

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

 

[Dana DeLouis]

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

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

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

(e-mail address removed)
Replace @mailinator.com with @tiscali.co.uk

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


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

(e-mail address removed)
Replace @mailinator.com with @tiscali.co.uk


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


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

(e-mail address removed)
Replace @mailinator.com with @tiscali.co.uk


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

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]

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

(e-mail address removed)
Replace @mailinator.com with @tiscali.co.uk

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

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

(e-mail address removed)
Replace @mailinator.com with @tiscali.co.uk

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


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

(e-mail address removed)
Replace @mailinator.com with @tiscali.co.uk


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


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

(e-mail address removed)
Replace @mailinator.com with @tiscali.co.uk


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

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]

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

(e-mail address removed)
Replace @mailinator.com with @tiscali.co.uk

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

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

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

(e-mail address removed)
Replace @mailinator.com with @tiscali.co.uk


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


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

(e-mail address removed)
Replace @mailinator.com with @tiscali.co.uk


"Rick Rothstein (MVP - VB)" <[email protected]>
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


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

(e-mail address removed)
Replace @mailinator.com with @tiscali.co.uk


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

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

 

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


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

(e-mail address removed)
Replace @mailinator.com with @tiscali.co.uk


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

(e-mail address removed)
Replace @mailinator.com with @tiscali.co.uk


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

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