Geometrical interpretation for the sum of factorial numbers
I am in need of a way to represent the sum
$1! + 2! + 3! + 4! = 1 + 2 + 6 + 24 = 33$
in a geometrical way. What I mean by this is that for example, the sum
$1^2 + 2^2 + 3^2 + 4^2 = 1 + 4 + 9 + 16 = 30$
can be represented geometrically as a pyramid with layers consisting of 1, 4, 9 and 16 pieces respectively a regular manner. Image from Wikipedia to illustrate the geometrical construction of the square numbers.
I have tried to find such a regular pattern to construct a geometrical shape from the factorial numbers, but to no avail. How could this be done?
Also, a follow up question: Is there a way to represent the factorial numbers up to an arbitrary number $n!$, instead of ending at $4!$ as stated in this question above? (less important, but interesting nonetheless)
Thanks in advance!
EDIT: The probably most important part is that the 1, 2, 6 and 24 are discrete and somewhat separated from each other, kind of like the different layers in the comparison between te sum of squares (see linked image above).
geometry factorial geometric-interpretation
asked Nov 30 '18 at 19:25
Sigfrid StjärnholmSigfrid Stjärnholm
434
add a comment |
I am in need of a way to represent the sum
$1! + 2! + 3! + 4! = 1 + 2 + 6 + 24 = 33$
in a geometrical way. What I mean by this is that for example, the sum
$1^2 + 2^2 + 3^2 + 4^2 = 1 + 4 + 9 + 16 = 30$
can be represented geometrically as a pyramid with layers consisting of 1, 4, 9 and 16 pieces respectively a regular manner. Image from Wikipedia to illustrate the geometrical construction of the square numbers.
I have tried to find such a regular pattern to construct a geometrical shape from the factorial numbers, but to no avail. How could this be done?
Also, a follow up question: Is there a way to represent the factorial numbers up to an arbitrary number $n!$, instead of ending at $4!$ as stated in this question above? (less important, but interesting nonetheless)
Thanks in advance!
EDIT: The probably most important part is that the 1, 2, 6 and 24 are discrete and somewhat separated from each other, kind of like the different layers in the comparison between te sum of squares (see linked image above).
geometry factorial geometric-interpretation
asked Nov 30 '18 at 19:25
Sigfrid StjärnholmSigfrid Stjärnholm
434
Why not use a triangle? Place a column of $1$s at the right. Starting from the second row, place a column of $2$s to the left of the column of $1$s. Repeat the column construction until you get to $n$ which should be a column of length one.
– John Douma
Nov 30 '18 at 19:45
@JohnDouma I am not quite sure how this would produce something connected to factorial sums? Thank you for your comment. Would I have first 4 1:s, then 3 2:s, 2 3:s, 1 4:s? That makes a sum of 20.
– Sigfrid Stjärnholm
Nov 30 '18 at 20:13
add a comment |
I am in need of a way to represent the sum
$1! + 2! + 3! + 4! = 1 + 2 + 6 + 24 = 33$
in a geometrical way. What I mean by this is that for example, the sum
$1^2 + 2^2 + 3^2 + 4^2 = 1 + 4 + 9 + 16 = 30$
can be represented geometrically as a pyramid with layers consisting of 1, 4, 9 and 16 pieces respectively a regular manner. Image from Wikipedia to illustrate the geometrical construction of the square numbers.
I have tried to find such a regular pattern to construct a geometrical shape from the factorial numbers, but to no avail. How could this be done?
Also, a follow up question: Is there a way to represent the factorial numbers up to an arbitrary number $n!$, instead of ending at $4!$ as stated in this question above? (less important, but interesting nonetheless)
Thanks in advance!
EDIT: The probably most important part is that the 1, 2, 6 and 24 are discrete and somewhat separated from each other, kind of like the different layers in the comparison between te sum of squares (see linked image above).
geometry factorial geometric-interpretation
asked Nov 30 '18 at 19:25
Sigfrid StjärnholmSigfrid Stjärnholm
434
I am in need of a way to represent the sum
$1! + 2! + 3! + 4! = 1 + 2 + 6 + 24 = 33$
in a geometrical way. What I mean by this is that for example, the sum
$1^2 + 2^2 + 3^2 + 4^2 = 1 + 4 + 9 + 16 = 30$
can be represented geometrically as a pyramid with layers consisting of 1, 4, 9 and 16 pieces respectively a regular manner. Image from Wikipedia to illustrate the geometrical construction of the square numbers.
I have tried to find such a regular pattern to construct a geometrical shape from the factorial numbers, but to no avail. How could this be done?
Also, a follow up question: Is there a way to represent the factorial numbers up to an arbitrary number $n!$, instead of ending at $4!$ as stated in this question above? (less important, but interesting nonetheless)
Thanks in advance!
EDIT: The probably most important part is that the 1, 2, 6 and 24 are discrete and somewhat separated from each other, kind of like the different layers in the comparison between te sum of squares (see linked image above).
geometry factorial geometric-interpretation
geometry factorial geometric-interpretation
asked Nov 30 '18 at 19:25
Sigfrid StjärnholmSigfrid Stjärnholm
434
asked Nov 30 '18 at 19:25
Sigfrid StjärnholmSigfrid Stjärnholm
434
edited Nov 30 '18 at 20:18
Sigfrid Stjärnholm
asked Nov 30 '18 at 19:25
Sigfrid StjärnholmSigfrid Stjärnholm
434
asked Nov 30 '18 at 19:25
Sigfrid StjärnholmSigfrid Stjärnholm
434
asked Nov 30 '18 at 19:25
Sigfrid StjärnholmSigfrid Stjärnholm
434
434
Why not use a triangle? Place a column of $1$s at the right. Starting from the second row, place a column of $2$s to the left of the column of $1$s. Repeat the column construction until you get to $n$ which should be a column of length one.
– John Douma
Nov 30 '18 at 19:45
@JohnDouma I am not quite sure how this would produce something connected to factorial sums? Thank you for your comment. Would I have first 4 1:s, then 3 2:s, 2 3:s, 1 4:s? That makes a sum of 20.
– Sigfrid Stjärnholm
Nov 30 '18 at 20:13
add a comment |
Why not use a triangle? Place a column of $1$s at the right. Starting from the second row, place a column of $2$s to the left of the column of $1$s. Repeat the column construction until you get to $n$ which should be a column of length one.
– John Douma
Nov 30 '18 at 19:45
@JohnDouma I am not quite sure how this would produce something connected to factorial sums? Thank you for your comment. Would I have first 4 1:s, then 3 2:s, 2 3:s, 1 4:s? That makes a sum of 20.
– Sigfrid Stjärnholm
Nov 30 '18 at 20:13
Why not use a triangle? Place a column of $1$s at the right. Starting from the second row, place a column of $2$s to the left of the column of $1$s. Repeat the column construction until you get to $n$ which should be a column of length one.
– John Douma
Nov 30 '18 at 19:45
Why not use a triangle? Place a column of $1$s at the right. Starting from the second row, place a column of $2$s to the left of the column of $1$s. Repeat the column construction until you get to $n$ which should be a column of length one.
– John Douma
Nov 30 '18 at 19:45
@JohnDouma I am not quite sure how this would produce something connected to factorial sums? Thank you for your comment. Would I have first 4 1:s, then 3 2:s, 2 3:s, 1 4:s? That makes a sum of 20.
– Sigfrid Stjärnholm
Nov 30 '18 at 20:13
@JohnDouma I am not quite sure how this would produce something connected to factorial sums? Thank you for your comment. Would I have first 4 1:s, then 3 2:s, 2 3:s, 1 4:s? That makes a sum of 20.
– Sigfrid Stjärnholm
Nov 30 '18 at 20:13
add a comment |
2 Answers
2
active
oldest
votes
The only idea I could come up with was counting the number of vertices in a tree graph that had branching ratios of $2, 3, 4, ldots$, so that the number of vertices on each level were $1!, 2!, 3!, 4!, ldots$.
If you need more layers ($n=6$), you might want a different layout (thanks to @HenrikSchumacher):
Radial embedding is particularly elegant and helpful too ($n=6$):
Perhaps such a three-dimensional representation would be appropriate:
The answer to your question "Also, a follow up question: Is there a way to represent the factorial numbers up to an arbitrary number n!, instead of ending at 4! as stated in this question above?" is:
$$sumlimits_{n=1}^k n! = (-1)^{k+1} Gamma (k+2) text{Subfactorial}[-k-2]-text{Subfactorial}[-1]-1$$
answered Nov 30 '18 at 19:30
David G. StorkDavid G. Stork
10.1k21332
OP wants to know geometrical interpretation. Is there any geometrical interpretation using this? Please explain :)
– tarit goswami
Nov 30 '18 at 19:35
In no way does this answer OP's question.
– MPW
Nov 30 '18 at 19:39
1
@MPW: "Also, a follow up question: Is there a way to represent the factorial numbers up to an arbitrary number n!, instead of ending at 4! as stated in this question above? "
– David G. Stork
Nov 30 '18 at 19:40
Thank you for your response! This gives me the sum of the factorial numbers up to a certain point k, and in my case I want k = 4. I see your argument that it would be hard to find such a geometric interpretation of factorial sums, but if we limit the value of k to 4, only considering 4!, is it not possible that for such small values of factorial there would exist a beautiful and simple interpretation as it did for the squares?
– Sigfrid Stjärnholm
Nov 30 '18 at 20:02
1
@SigfridStjärnholm: Well, not precisely a fractal because the branching ratio changes at each level... hence the graph is not self-similar.
– David G. Stork
Nov 30 '18 at 22:46
|
show 2 more comments
Polynomial functions of degree $d$ can be represented in a space of $d$ dimensions, using segments, squares, cubes then hypercubes. For example, the square pyramidal numbers can be sketeched in 3D as a stack of squares.
This does not generalize to factorials as they are of "unbounded degree" and would require an unbounded number of dimensions.
Even if you find a trick to limit the number of dimensions, the value of the numbers quickly becomes unmanageable ($10!=3628800$).
answered Nov 30 '18 at 19:48
Yves DaoustYves Daoust
124k671222
Ah, i see! Would there however be some sort of way to represent just up to 4!, if ignoring the general case?
– Sigfrid Stjärnholm
Nov 30 '18 at 19:59
add a comment |
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2 Answers
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2 Answers
2
active
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oldest
votes
The only idea I could come up with was counting the number of vertices in a tree graph that had branching ratios of $2, 3, 4, ldots$, so that the number of vertices on each level were $1!, 2!, 3!, 4!, ldots$.
If you need more layers ($n=6$), you might want a different layout (thanks to @HenrikSchumacher):
Radial embedding is particularly elegant and helpful too ($n=6$):
Perhaps such a three-dimensional representation would be appropriate:
The answer to your question "Also, a follow up question: Is there a way to represent the factorial numbers up to an arbitrary number n!, instead of ending at 4! as stated in this question above?" is:
$$sumlimits_{n=1}^k n! = (-1)^{k+1} Gamma (k+2) text{Subfactorial}[-k-2]-text{Subfactorial}[-1]-1$$
answered Nov 30 '18 at 19:30
David G. StorkDavid G. Stork
10.1k21332
OP wants to know geometrical interpretation. Is there any geometrical interpretation using this? Please explain :)
– tarit goswami
Nov 30 '18 at 19:35
In no way does this answer OP's question.
– MPW
Nov 30 '18 at 19:39
1
@MPW: "Also, a follow up question: Is there a way to represent the factorial numbers up to an arbitrary number n!, instead of ending at 4! as stated in this question above? "
– David G. Stork
Nov 30 '18 at 19:40
Thank you for your response! This gives me the sum of the factorial numbers up to a certain point k, and in my case I want k = 4. I see your argument that it would be hard to find such a geometric interpretation of factorial sums, but if we limit the value of k to 4, only considering 4!, is it not possible that for such small values of factorial there would exist a beautiful and simple interpretation as it did for the squares?
– Sigfrid Stjärnholm
Nov 30 '18 at 20:02
1
@SigfridStjärnholm: Well, not precisely a fractal because the branching ratio changes at each level... hence the graph is not self-similar.
– David G. Stork
Nov 30 '18 at 22:46
|
show 2 more comments
The only idea I could come up with was counting the number of vertices in a tree graph that had branching ratios of $2, 3, 4, ldots$, so that the number of vertices on each level were $1!, 2!, 3!, 4!, ldots$.
If you need more layers ($n=6$), you might want a different layout (thanks to @HenrikSchumacher):
Radial embedding is particularly elegant and helpful too ($n=6$):
Perhaps such a three-dimensional representation would be appropriate:
The answer to your question "Also, a follow up question: Is there a way to represent the factorial numbers up to an arbitrary number n!, instead of ending at 4! as stated in this question above?" is:
$$sumlimits_{n=1}^k n! = (-1)^{k+1} Gamma (k+2) text{Subfactorial}[-k-2]-text{Subfactorial}[-1]-1$$
answered Nov 30 '18 at 19:30
David G. StorkDavid G. Stork
10.1k21332
OP wants to know geometrical interpretation. Is there any geometrical interpretation using this? Please explain :)
– tarit goswami
Nov 30 '18 at 19:35
In no way does this answer OP's question.
– MPW
Nov 30 '18 at 19:39
1
@MPW: "Also, a follow up question: Is there a way to represent the factorial numbers up to an arbitrary number n!, instead of ending at 4! as stated in this question above? "
– David G. Stork
Nov 30 '18 at 19:40
Thank you for your response! This gives me the sum of the factorial numbers up to a certain point k, and in my case I want k = 4. I see your argument that it would be hard to find such a geometric interpretation of factorial sums, but if we limit the value of k to 4, only considering 4!, is it not possible that for such small values of factorial there would exist a beautiful and simple interpretation as it did for the squares?
– Sigfrid Stjärnholm
Nov 30 '18 at 20:02
1
@SigfridStjärnholm: Well, not precisely a fractal because the branching ratio changes at each level... hence the graph is not self-similar.
– David G. Stork
Nov 30 '18 at 22:46
|
show 2 more comments
The only idea I could come up with was counting the number of vertices in a tree graph that had branching ratios of $2, 3, 4, ldots$, so that the number of vertices on each level were $1!, 2!, 3!, 4!, ldots$.
If you need more layers ($n=6$), you might want a different layout (thanks to @HenrikSchumacher):
Radial embedding is particularly elegant and helpful too ($n=6$):
Perhaps such a three-dimensional representation would be appropriate:
The answer to your question "Also, a follow up question: Is there a way to represent the factorial numbers up to an arbitrary number n!, instead of ending at 4! as stated in this question above?" is:
$$sumlimits_{n=1}^k n! = (-1)^{k+1} Gamma (k+2) text{Subfactorial}[-k-2]-text{Subfactorial}[-1]-1$$
answered Nov 30 '18 at 19:30
David G. StorkDavid G. Stork
10.1k21332
The only idea I could come up with was counting the number of vertices in a tree graph that had branching ratios of $2, 3, 4, ldots$, so that the number of vertices on each level were $1!, 2!, 3!, 4!, ldots$.
If you need more layers ($n=6$), you might want a different layout (thanks to @HenrikSchumacher):
Radial embedding is particularly elegant and helpful too ($n=6$):
Perhaps such a three-dimensional representation would be appropriate:
The answer to your question "Also, a follow up question: Is there a way to represent the factorial numbers up to an arbitrary number n!, instead of ending at 4! as stated in this question above?" is:
$$sumlimits_{n=1}^k n! = (-1)^{k+1} Gamma (k+2) text{Subfactorial}[-k-2]-text{Subfactorial}[-1]-1$$
answered Nov 30 '18 at 19:30
David G. StorkDavid G. Stork
10.1k21332
edited Dec 1 '18 at 1:16
answered Nov 30 '18 at 19:30
David G. StorkDavid G. Stork
10.1k21332
answered Nov 30 '18 at 19:30
David G. StorkDavid G. Stork
10.1k21332
answered Nov 30 '18 at 19:30
David G. StorkDavid G. Stork
10.1k21332
10.1k21332
OP wants to know geometrical interpretation. Is there any geometrical interpretation using this? Please explain :)
– tarit goswami
Nov 30 '18 at 19:35
In no way does this answer OP's question.
– MPW
Nov 30 '18 at 19:39
1
@MPW: "Also, a follow up question: Is there a way to represent the factorial numbers up to an arbitrary number n!, instead of ending at 4! as stated in this question above? "
– David G. Stork
Nov 30 '18 at 19:40
Thank you for your response! This gives me the sum of the factorial numbers up to a certain point k, and in my case I want k = 4. I see your argument that it would be hard to find such a geometric interpretation of factorial sums, but if we limit the value of k to 4, only considering 4!, is it not possible that for such small values of factorial there would exist a beautiful and simple interpretation as it did for the squares?
– Sigfrid Stjärnholm
Nov 30 '18 at 20:02
1
@SigfridStjärnholm: Well, not precisely a fractal because the branching ratio changes at each level... hence the graph is not self-similar.
– David G. Stork
Nov 30 '18 at 22:46
|
show 2 more comments
OP wants to know geometrical interpretation. Is there any geometrical interpretation using this? Please explain :)
– tarit goswami
Nov 30 '18 at 19:35
In no way does this answer OP's question.
– MPW
Nov 30 '18 at 19:39
1
@MPW: "Also, a follow up question: Is there a way to represent the factorial numbers up to an arbitrary number n!, instead of ending at 4! as stated in this question above? "
– David G. Stork
Nov 30 '18 at 19:40
Thank you for your response! This gives me the sum of the factorial numbers up to a certain point k, and in my case I want k = 4. I see your argument that it would be hard to find such a geometric interpretation of factorial sums, but if we limit the value of k to 4, only considering 4!, is it not possible that for such small values of factorial there would exist a beautiful and simple interpretation as it did for the squares?
– Sigfrid Stjärnholm
Nov 30 '18 at 20:02
1
@SigfridStjärnholm: Well, not precisely a fractal because the branching ratio changes at each level... hence the graph is not self-similar.
– David G. Stork
Nov 30 '18 at 22:46
OP wants to know geometrical interpretation. Is there any geometrical interpretation using this? Please explain :)
– tarit goswami
Nov 30 '18 at 19:35
OP wants to know geometrical interpretation. Is there any geometrical interpretation using this? Please explain :)
– tarit goswami
Nov 30 '18 at 19:35
In no way does this answer OP's question.
– MPW
Nov 30 '18 at 19:39
In no way does this answer OP's question.
– MPW
Nov 30 '18 at 19:39
1
1
@MPW: "Also, a follow up question: Is there a way to represent the factorial numbers up to an arbitrary number n!, instead of ending at 4! as stated in this question above? "
– David G. Stork
Nov 30 '18 at 19:40
@MPW: "Also, a follow up question: Is there a way to represent the factorial numbers up to an arbitrary number n!, instead of ending at 4! as stated in this question above? "
– David G. Stork
Nov 30 '18 at 19:40
Thank you for your response! This gives me the sum of the factorial numbers up to a certain point k, and in my case I want k = 4. I see your argument that it would be hard to find such a geometric interpretation of factorial sums, but if we limit the value of k to 4, only considering 4!, is it not possible that for such small values of factorial there would exist a beautiful and simple interpretation as it did for the squares?
– Sigfrid Stjärnholm
Nov 30 '18 at 20:02
Thank you for your response! This gives me the sum of the factorial numbers up to a certain point k, and in my case I want k = 4. I see your argument that it would be hard to find such a geometric interpretation of factorial sums, but if we limit the value of k to 4, only considering 4!, is it not possible that for such small values of factorial there would exist a beautiful and simple interpretation as it did for the squares?
– Sigfrid Stjärnholm
Nov 30 '18 at 20:02
1
1
@SigfridStjärnholm: Well, not precisely a fractal because the branching ratio changes at each level... hence the graph is not self-similar.
– David G. Stork
Nov 30 '18 at 22:46
@SigfridStjärnholm: Well, not precisely a fractal because the branching ratio changes at each level... hence the graph is not self-similar.
– David G. Stork
Nov 30 '18 at 22:46
|
show 2 more comments
Polynomial functions of degree $d$ can be represented in a space of $d$ dimensions, using segments, squares, cubes then hypercubes. For example, the square pyramidal numbers can be sketeched in 3D as a stack of squares.
This does not generalize to factorials as they are of "unbounded degree" and would require an unbounded number of dimensions.
Even if you find a trick to limit the number of dimensions, the value of the numbers quickly becomes unmanageable ($10!=3628800$).
answered Nov 30 '18 at 19:48
Yves DaoustYves Daoust
124k671222
Ah, i see! Would there however be some sort of way to represent just up to 4!, if ignoring the general case?
– Sigfrid Stjärnholm
Nov 30 '18 at 19:59
add a comment |
Polynomial functions of degree $d$ can be represented in a space of $d$ dimensions, using segments, squares, cubes then hypercubes. For example, the square pyramidal numbers can be sketeched in 3D as a stack of squares.
This does not generalize to factorials as they are of "unbounded degree" and would require an unbounded number of dimensions.
Even if you find a trick to limit the number of dimensions, the value of the numbers quickly becomes unmanageable ($10!=3628800$).
answered Nov 30 '18 at 19:48
Yves DaoustYves Daoust
124k671222
Ah, i see! Would there however be some sort of way to represent just up to 4!, if ignoring the general case?
– Sigfrid Stjärnholm
Nov 30 '18 at 19:59
add a comment |
Polynomial functions of degree $d$ can be represented in a space of $d$ dimensions, using segments, squares, cubes then hypercubes. For example, the square pyramidal numbers can be sketeched in 3D as a stack of squares.
This does not generalize to factorials as they are of "unbounded degree" and would require an unbounded number of dimensions.
Even if you find a trick to limit the number of dimensions, the value of the numbers quickly becomes unmanageable ($10!=3628800$).
answered Nov 30 '18 at 19:48
Yves DaoustYves Daoust
124k671222
Polynomial functions of degree $d$ can be represented in a space of $d$ dimensions, using segments, squares, cubes then hypercubes. For example, the square pyramidal numbers can be sketeched in 3D as a stack of squares.
This does not generalize to factorials as they are of "unbounded degree" and would require an unbounded number of dimensions.
Even if you find a trick to limit the number of dimensions, the value of the numbers quickly becomes unmanageable ($10!=3628800$).
answered Nov 30 '18 at 19:48
Yves DaoustYves Daoust
124k671222
answered Nov 30 '18 at 19:48
Yves DaoustYves Daoust
124k671222
answered Nov 30 '18 at 19:48
Yves DaoustYves Daoust
124k671222
answered Nov 30 '18 at 19:48
Yves DaoustYves Daoust
124k671222
124k671222
Ah, i see! Would there however be some sort of way to represent just up to 4!, if ignoring the general case?
– Sigfrid Stjärnholm
Nov 30 '18 at 19:59
add a comment |
Ah, i see! Would there however be some sort of way to represent just up to 4!, if ignoring the general case?
– Sigfrid Stjärnholm
Nov 30 '18 at 19:59
Ah, i see! Would there however be some sort of way to represent just up to 4!, if ignoring the general case?
– Sigfrid Stjärnholm
Nov 30 '18 at 19:59
Ah, i see! Would there however be some sort of way to represent just up to 4!, if ignoring the general case?
– Sigfrid Stjärnholm
Nov 30 '18 at 19:59
add a comment |
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Why not use a triangle? Place a column of $1$s at the right. Starting from the second row, place a column of $2$s to the left of the column of $1$s. Repeat the column construction until you get to $n$ which should be a column of length one.
– John Douma
Nov 30 '18 at 19:45
@JohnDouma I am not quite sure how this would produce something connected to factorial sums? Thank you for your comment. Would I have first 4 1:s, then 3 2:s, 2 3:s, 1 4:s? That makes a sum of 20.
– Sigfrid Stjärnholm
Nov 30 '18 at 20:13