Mixing
Geometrical interpretation of $(sum_{k=1}^n k)^2=sum_{k=1}^n k^3$
$begingroup$
Using induction it is straight forward to show
$$left(sum_{k=1}^n kright)^2=sum_{k=1}^n k^3.$$
But is there also a geometrical interpretation that "proves" this fact? By just looking at those formulas I don't see why they should be equal.
summation induction
asked May 30 '15 at 10:00
principal-ideal-domainprincipal-ideal-domain
2,756622
$endgroup$
add a comment |
$begingroup$
Using induction it is straight forward to show
$$left(sum_{k=1}^n kright)^2=sum_{k=1}^n k^3.$$
But is there also a geometrical interpretation that "proves" this fact? By just looking at those formulas I don't see why they should be equal.
summation induction
asked May 30 '15 at 10:00
principal-ideal-domainprincipal-ideal-domain
2,756622
$endgroup$
$begingroup$
See also math.stackexchange.com/q/320985/9754
$endgroup$
– davidlowryduda♦
Aug 9 '16 at 16:41
add a comment |
$begingroup$
Using induction it is straight forward to show
$$left(sum_{k=1}^n kright)^2=sum_{k=1}^n k^3.$$
But is there also a geometrical interpretation that "proves" this fact? By just looking at those formulas I don't see why they should be equal.
summation induction
asked May 30 '15 at 10:00
principal-ideal-domainprincipal-ideal-domain
2,756622
$endgroup$
Using induction it is straight forward to show
$$left(sum_{k=1}^n kright)^2=sum_{k=1}^n k^3.$$
But is there also a geometrical interpretation that "proves" this fact? By just looking at those formulas I don't see why they should be equal.
summation induction
summation induction
asked May 30 '15 at 10:00
principal-ideal-domainprincipal-ideal-domain
2,756622
asked May 30 '15 at 10:00
principal-ideal-domainprincipal-ideal-domain
2,756622
asked May 30 '15 at 10:00
principal-ideal-domainprincipal-ideal-domain
2,756622
asked May 30 '15 at 10:00
principal-ideal-domainprincipal-ideal-domain
2,756622
asked May 30 '15 at 10:00
principal-ideal-domainprincipal-ideal-domain
2,756622
2,756622
$begingroup$
See also math.stackexchange.com/q/320985/9754
$endgroup$
– davidlowryduda♦
Aug 9 '16 at 16:41
add a comment |
$begingroup$
See also math.stackexchange.com/q/320985/9754
$endgroup$
– davidlowryduda♦
Aug 9 '16 at 16:41
$begingroup$
See also math.stackexchange.com/q/320985/9754
$endgroup$
– davidlowryduda♦
Aug 9 '16 at 16:41
$begingroup$
See also math.stackexchange.com/q/320985/9754
$endgroup$
– davidlowryduda♦
Aug 9 '16 at 16:41
add a comment |
5 Answers
5
active
oldest
votes
$begingroup$
The clearest proof I've seen is this one, which comes from here: you just have to stare at it for a few seconds to see how it works. (It's a variant of the other proofs, of course, but actually has cubes in it.)
answered May 30 '15 at 10:18
ChappersChappers
56k74295
$endgroup$
add a comment |
$begingroup$
There is this picture. Here, they represent $x^3$ as $x$ squares of side length $x$. The big square is the sum of all numbers up to $x$.
answered May 30 '15 at 10:05
Linus S.Linus S.
1,823518
$endgroup$
3
$begingroup$
Who are they? Add a reference.
$endgroup$
– lhf
May 30 '15 at 10:13
add a comment |
$begingroup$
This identity is sometimes called Nicomachus's theorem. If you type this in google, you'll recieve numerous pictures.
1.
2.
3.
answered May 30 '15 at 10:54
aGeraGer
821823
$endgroup$
add a comment |
$begingroup$
page 86 ,85
book :proof without words
author : roger nelsen
you can find here many kind of this proof
answered May 30 '15 at 10:28
KhosrotashKhosrotash
16.7k12461
$endgroup$
add a comment |
$begingroup$
An engineering style fourier-approach. You may not consider this to be very intuitively "geometric", but I thought it could be interesting however.
Consider the time-signal $[1,2,...,k]$ (A linear function or "triangle wave"). The LHS is the square of the DC component of the signal in the temporal/spatial domain which is straight forward to calculate. The RHS is the iterated convolution of the fourier transform of [1,2,3,...,k] three times and then DC component of that.
answered May 30 '15 at 11:48
mathreadlermathreadler
15.4k72263
$endgroup$
$begingroup$
Could you please take a look at this
$endgroup$
– Iuʇǝƃɹɐʇoɹ
Sep 18 '15 at 1:09
add a comment |
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5 Answers
5
active
oldest
votes
5 Answers
5
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
The clearest proof I've seen is this one, which comes from here: you just have to stare at it for a few seconds to see how it works. (It's a variant of the other proofs, of course, but actually has cubes in it.)
answered May 30 '15 at 10:18
ChappersChappers
56k74295
$endgroup$
add a comment |
$begingroup$
The clearest proof I've seen is this one, which comes from here: you just have to stare at it for a few seconds to see how it works. (It's a variant of the other proofs, of course, but actually has cubes in it.)
answered May 30 '15 at 10:18
ChappersChappers
56k74295
$endgroup$
add a comment |
$begingroup$
The clearest proof I've seen is this one, which comes from here: you just have to stare at it for a few seconds to see how it works. (It's a variant of the other proofs, of course, but actually has cubes in it.)
answered May 30 '15 at 10:18
ChappersChappers
56k74295
$endgroup$
The clearest proof I've seen is this one, which comes from here: you just have to stare at it for a few seconds to see how it works. (It's a variant of the other proofs, of course, but actually has cubes in it.)
answered May 30 '15 at 10:18
ChappersChappers
56k74295
answered May 30 '15 at 10:18
ChappersChappers
56k74295
answered May 30 '15 at 10:18
ChappersChappers
56k74295
answered May 30 '15 at 10:18
ChappersChappers
56k74295
56k74295
add a comment |
add a comment |
$begingroup$
There is this picture. Here, they represent $x^3$ as $x$ squares of side length $x$. The big square is the sum of all numbers up to $x$.
answered May 30 '15 at 10:05
Linus S.Linus S.
1,823518
$endgroup$
3
$begingroup$
Who are they? Add a reference.
$endgroup$
– lhf
May 30 '15 at 10:13
add a comment |
$begingroup$
There is this picture. Here, they represent $x^3$ as $x$ squares of side length $x$. The big square is the sum of all numbers up to $x$.
answered May 30 '15 at 10:05
Linus S.Linus S.
1,823518
$endgroup$
3
$begingroup$
Who are they? Add a reference.
$endgroup$
– lhf
May 30 '15 at 10:13
add a comment |
$begingroup$
There is this picture. Here, they represent $x^3$ as $x$ squares of side length $x$. The big square is the sum of all numbers up to $x$.
answered May 30 '15 at 10:05
Linus S.Linus S.
1,823518
$endgroup$
There is this picture. Here, they represent $x^3$ as $x$ squares of side length $x$. The big square is the sum of all numbers up to $x$.
answered May 30 '15 at 10:05
Linus S.Linus S.
1,823518
answered May 30 '15 at 10:05
Linus S.Linus S.
1,823518
answered May 30 '15 at 10:05
Linus S.Linus S.
1,823518
answered May 30 '15 at 10:05
Linus S.Linus S.
1,823518
1,823518
3
$begingroup$
Who are they? Add a reference.
$endgroup$
– lhf
May 30 '15 at 10:13
add a comment |
3
$begingroup$
Who are they? Add a reference.
$endgroup$
– lhf
May 30 '15 at 10:13
3
3
$begingroup$
Who are they? Add a reference.
$endgroup$
– lhf
May 30 '15 at 10:13
$begingroup$
Who are they? Add a reference.
$endgroup$
– lhf
May 30 '15 at 10:13
add a comment |
$begingroup$
This identity is sometimes called Nicomachus's theorem. If you type this in google, you'll recieve numerous pictures.
1.
2.
3.
answered May 30 '15 at 10:54
aGeraGer
821823
$endgroup$
add a comment |
$begingroup$
This identity is sometimes called Nicomachus's theorem. If you type this in google, you'll recieve numerous pictures.
1.
2.
3.
answered May 30 '15 at 10:54
aGeraGer
821823
$endgroup$
add a comment |
$begingroup$
This identity is sometimes called Nicomachus's theorem. If you type this in google, you'll recieve numerous pictures.
1.
2.
3.
answered May 30 '15 at 10:54
aGeraGer
821823
$endgroup$
This identity is sometimes called Nicomachus's theorem. If you type this in google, you'll recieve numerous pictures.
1.
2.
3.
answered May 30 '15 at 10:54
aGeraGer
821823
answered May 30 '15 at 10:54
aGeraGer
821823
answered May 30 '15 at 10:54
aGeraGer
821823
answered May 30 '15 at 10:54
aGeraGer
821823
821823
add a comment |
add a comment |
$begingroup$
page 86 ,85
book :proof without words
author : roger nelsen
you can find here many kind of this proof
answered May 30 '15 at 10:28
KhosrotashKhosrotash
16.7k12461
$endgroup$
add a comment |
$begingroup$
page 86 ,85
book :proof without words
author : roger nelsen
you can find here many kind of this proof
answered May 30 '15 at 10:28
KhosrotashKhosrotash
16.7k12461
$endgroup$
add a comment |
$begingroup$
page 86 ,85
book :proof without words
author : roger nelsen
you can find here many kind of this proof
answered May 30 '15 at 10:28
KhosrotashKhosrotash
16.7k12461
$endgroup$
page 86 ,85
book :proof without words
author : roger nelsen
you can find here many kind of this proof
answered May 30 '15 at 10:28
KhosrotashKhosrotash
16.7k12461
answered May 30 '15 at 10:28
KhosrotashKhosrotash
16.7k12461
answered May 30 '15 at 10:28
KhosrotashKhosrotash
16.7k12461
answered May 30 '15 at 10:28
KhosrotashKhosrotash
16.7k12461
16.7k12461
add a comment |
add a comment |
$begingroup$
An engineering style fourier-approach. You may not consider this to be very intuitively "geometric", but I thought it could be interesting however.
Consider the time-signal $[1,2,...,k]$ (A linear function or "triangle wave"). The LHS is the square of the DC component of the signal in the temporal/spatial domain which is straight forward to calculate. The RHS is the iterated convolution of the fourier transform of [1,2,3,...,k] three times and then DC component of that.
answered May 30 '15 at 11:48
mathreadlermathreadler
15.4k72263
$endgroup$
$begingroup$
Could you please take a look at this
$endgroup$
– Iuʇǝƃɹɐʇoɹ
Sep 18 '15 at 1:09
add a comment |
$begingroup$
An engineering style fourier-approach. You may not consider this to be very intuitively "geometric", but I thought it could be interesting however.
Consider the time-signal $[1,2,...,k]$ (A linear function or "triangle wave"). The LHS is the square of the DC component of the signal in the temporal/spatial domain which is straight forward to calculate. The RHS is the iterated convolution of the fourier transform of [1,2,3,...,k] three times and then DC component of that.
answered May 30 '15 at 11:48
mathreadlermathreadler
15.4k72263
$endgroup$
$begingroup$
Could you please take a look at this
$endgroup$
– Iuʇǝƃɹɐʇoɹ
Sep 18 '15 at 1:09
add a comment |
$begingroup$
An engineering style fourier-approach. You may not consider this to be very intuitively "geometric", but I thought it could be interesting however.
Consider the time-signal $[1,2,...,k]$ (A linear function or "triangle wave"). The LHS is the square of the DC component of the signal in the temporal/spatial domain which is straight forward to calculate. The RHS is the iterated convolution of the fourier transform of [1,2,3,...,k] three times and then DC component of that.
answered May 30 '15 at 11:48
mathreadlermathreadler
15.4k72263
$endgroup$
An engineering style fourier-approach. You may not consider this to be very intuitively "geometric", but I thought it could be interesting however.
Consider the time-signal $[1,2,...,k]$ (A linear function or "triangle wave"). The LHS is the square of the DC component of the signal in the temporal/spatial domain which is straight forward to calculate. The RHS is the iterated convolution of the fourier transform of [1,2,3,...,k] three times and then DC component of that.
answered May 30 '15 at 11:48
mathreadlermathreadler
15.4k72263
answered May 30 '15 at 11:48
mathreadlermathreadler
15.4k72263
answered May 30 '15 at 11:48
mathreadlermathreadler
15.4k72263
answered May 30 '15 at 11:48
mathreadlermathreadler
15.4k72263
15.4k72263
$begingroup$
Could you please take a look at this
$endgroup$
– Iuʇǝƃɹɐʇoɹ
Sep 18 '15 at 1:09
add a comment |
$begingroup$
Could you please take a look at this
$endgroup$
– Iuʇǝƃɹɐʇoɹ
Sep 18 '15 at 1:09
$begingroup$
Could you please take a look at this
$endgroup$
– Iuʇǝƃɹɐʇoɹ
Sep 18 '15 at 1:09
$begingroup$
Could you please take a look at this
$endgroup$
– Iuʇǝƃɹɐʇoɹ
Sep 18 '15 at 1:09
add a comment |
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See also math.stackexchange.com/q/320985/9754
$endgroup$
– davidlowryduda♦
Aug 9 '16 at 16:41