spring-of-mathematics:

Image 1: Multiplication of the integers modulo 15 - A beautiful illustration showing the emergence of primes and symmetry of multiplication. The colors were chosen to start blue at 1 (cold) and fade to red at n (hot). White is used for zero (frozen), because it communicates the most information about prime factorization.Image 2: Multiplication of the integers modulo 512.
The interactive version can be found here at ℤn Multiplication Visualizer βeta by Jared Deckard.Shared at: Math.stackexchange.com743542/1
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spring-of-mathematics:

Image 1: Multiplication of the integers modulo 15 - A beautiful illustration showing the emergence of primes and symmetry of multiplication. The colors were chosen to start blue at 1 (cold) and fade to red at n (hot). White is used for zero (frozen), because it communicates the most information about prime factorization.Image 2: Multiplication of the integers modulo 512.
The interactive version can be found here at ℤn Multiplication Visualizer βeta by Jared Deckard.Shared at: Math.stackexchange.com743542/1
Zoom Info

spring-of-mathematics:

Image 1: Multiplication of the integers modulo 15 - A beautiful illustration showing the emergence of primes and symmetry of multiplication. The colors were chosen to start blue at 1 (cold) and fade to red at n (hot). White is used for zero (frozen), because it communicates the most information about prime factorization.
Image 2: Multiplication of the integers modulo 512.

The interactive version can be found here at ℤn Multiplication Visualizer βeta by Jared Deckard.
Shared at: Math.stackexchange.com743542/1

spring-of-mathematics:

How many non-overlapping triangles can be formed in an arrangement of k lines?
The Kobon triangle problem is an unsolved problem in combinatorial geometry first stated by Kobon Fujimura. The problem asks for the largest number N(k) of nonoverlapping triangles whose sides lie on an arrangement of k lines. Variations of the problem consider the projective plane rather than the Euclidean plane, and require that the triangles not be crossed by any other lines of the arrangement.
The problem of finding a formula for the maximum number of triangles as a function of the number of lines and find an analytic expression for the nth term appears to be extremely difficult.
See more at: http://mathworld.wolfram.com/KobonTriangle.html & http://en.wikipedia.org/wiki/Kobon_triangle_problem.
Image shared at: http://www.ilemaths.net & http://en.wikipedia.org.
Zoom Info
spring-of-mathematics:

How many non-overlapping triangles can be formed in an arrangement of k lines?
The Kobon triangle problem is an unsolved problem in combinatorial geometry first stated by Kobon Fujimura. The problem asks for the largest number N(k) of nonoverlapping triangles whose sides lie on an arrangement of k lines. Variations of the problem consider the projective plane rather than the Euclidean plane, and require that the triangles not be crossed by any other lines of the arrangement.
The problem of finding a formula for the maximum number of triangles as a function of the number of lines and find an analytic expression for the nth term appears to be extremely difficult.
See more at: http://mathworld.wolfram.com/KobonTriangle.html & http://en.wikipedia.org/wiki/Kobon_triangle_problem.
Image shared at: http://www.ilemaths.net & http://en.wikipedia.org.
Zoom Info
spring-of-mathematics:

How many non-overlapping triangles can be formed in an arrangement of k lines?
The Kobon triangle problem is an unsolved problem in combinatorial geometry first stated by Kobon Fujimura. The problem asks for the largest number N(k) of nonoverlapping triangles whose sides lie on an arrangement of k lines. Variations of the problem consider the projective plane rather than the Euclidean plane, and require that the triangles not be crossed by any other lines of the arrangement.
The problem of finding a formula for the maximum number of triangles as a function of the number of lines and find an analytic expression for the nth term appears to be extremely difficult.
See more at: http://mathworld.wolfram.com/KobonTriangle.html & http://en.wikipedia.org/wiki/Kobon_triangle_problem.
Image shared at: http://www.ilemaths.net & http://en.wikipedia.org.
Zoom Info

spring-of-mathematics:

How many non-overlapping triangles can be formed in an arrangement of k lines?

The Kobon triangle problem is an unsolved problem in combinatorial geometry first stated by Kobon Fujimura. The problem asks for the largest number N(k) of nonoverlapping triangles whose sides lie on an arrangement of k lines. Variations of the problem consider the projective plane rather than the Euclidean plane, and require that the triangles not be crossed by any other lines of the arrangement.

The problem of finding a formula for the maximum number of triangles as a function of the number of lines and find an analytic expression for the nth term appears to be extremely difficult.

See more at: http://mathworld.wolfram.com/KobonTriangle.html & http://en.wikipedia.org/wiki/Kobon_triangle_problem.

Image shared at: http://www.ilemaths.net & http://en.wikipedia.org.

spring-of-mathematics:

Fourier Transform
In the first frames of the animation, a function f is resolved into Fourier series: a linear combination of sines and cosines (in blue). The component frequencies of these sines and cosines spread across the frequency spectrum, are represented as peaks in the frequency domain (actually Dirac delta functions, shown in the last frames of the animation). The frequency domain representation of the function, f^ (f-hat), is the collection of these peaks at the frequencies that appear in this resolution of the function.
Image (Animated gif): The Fourier transform takes an input function f (in red) in the “time domain” and converts it into a new function f-hat (in blue) in the “frequency domain”. In other words, the original function can be thought of as being “amplitude given time”, and the Fourier transform of the function is “amplitude given frequency”.Shown here, a simple 6-component approximation of the square wave is decomposed (exactly, for simplicity) into 6 sine waves. These component frequencies show as very sharp peaks in the frequency domain of the function, shown as the blue graph. In practice, these peaks are never that sharp. That would require infinite precision.
Image (gif animation) by Lucas Vieira Barbosa - http://1ucasvb.tumblr.com.
See more: Replicate the Fourier transform time-frequency domains correspondence illustration using TikZ (at here).
Zoom Info
spring-of-mathematics:

Fourier Transform
In the first frames of the animation, a function f is resolved into Fourier series: a linear combination of sines and cosines (in blue). The component frequencies of these sines and cosines spread across the frequency spectrum, are represented as peaks in the frequency domain (actually Dirac delta functions, shown in the last frames of the animation). The frequency domain representation of the function, f^ (f-hat), is the collection of these peaks at the frequencies that appear in this resolution of the function.
Image (Animated gif): The Fourier transform takes an input function f (in red) in the “time domain” and converts it into a new function f-hat (in blue) in the “frequency domain”. In other words, the original function can be thought of as being “amplitude given time”, and the Fourier transform of the function is “amplitude given frequency”.Shown here, a simple 6-component approximation of the square wave is decomposed (exactly, for simplicity) into 6 sine waves. These component frequencies show as very sharp peaks in the frequency domain of the function, shown as the blue graph. In practice, these peaks are never that sharp. That would require infinite precision.
Image (gif animation) by Lucas Vieira Barbosa - http://1ucasvb.tumblr.com.
See more: Replicate the Fourier transform time-frequency domains correspondence illustration using TikZ (at here).
Zoom Info
spring-of-mathematics:

Fourier Transform
In the first frames of the animation, a function f is resolved into Fourier series: a linear combination of sines and cosines (in blue). The component frequencies of these sines and cosines spread across the frequency spectrum, are represented as peaks in the frequency domain (actually Dirac delta functions, shown in the last frames of the animation). The frequency domain representation of the function, f^ (f-hat), is the collection of these peaks at the frequencies that appear in this resolution of the function.
Image (Animated gif): The Fourier transform takes an input function f (in red) in the “time domain” and converts it into a new function f-hat (in blue) in the “frequency domain”. In other words, the original function can be thought of as being “amplitude given time”, and the Fourier transform of the function is “amplitude given frequency”.Shown here, a simple 6-component approximation of the square wave is decomposed (exactly, for simplicity) into 6 sine waves. These component frequencies show as very sharp peaks in the frequency domain of the function, shown as the blue graph. In practice, these peaks are never that sharp. That would require infinite precision.
Image (gif animation) by Lucas Vieira Barbosa - http://1ucasvb.tumblr.com.
See more: Replicate the Fourier transform time-frequency domains correspondence illustration using TikZ (at here).
Zoom Info
spring-of-mathematics:

Fourier Transform
In the first frames of the animation, a function f is resolved into Fourier series: a linear combination of sines and cosines (in blue). The component frequencies of these sines and cosines spread across the frequency spectrum, are represented as peaks in the frequency domain (actually Dirac delta functions, shown in the last frames of the animation). The frequency domain representation of the function, f^ (f-hat), is the collection of these peaks at the frequencies that appear in this resolution of the function.
Image (Animated gif): The Fourier transform takes an input function f (in red) in the “time domain” and converts it into a new function f-hat (in blue) in the “frequency domain”. In other words, the original function can be thought of as being “amplitude given time”, and the Fourier transform of the function is “amplitude given frequency”.Shown here, a simple 6-component approximation of the square wave is decomposed (exactly, for simplicity) into 6 sine waves. These component frequencies show as very sharp peaks in the frequency domain of the function, shown as the blue graph. In practice, these peaks are never that sharp. That would require infinite precision.
Image (gif animation) by Lucas Vieira Barbosa - http://1ucasvb.tumblr.com.
See more: Replicate the Fourier transform time-frequency domains correspondence illustration using TikZ (at here).
Zoom Info

spring-of-mathematics:

Fourier Transform

In the first frames of the animation, a function f is resolved into Fourier series: a linear combination of sines and cosines (in blue). The component frequencies of these sines and cosines spread across the frequency spectrum, are represented as peaks in the frequency domain (actually Dirac delta functions, shown in the last frames of the animation). The frequency domain representation of the function, f^ (f-hat), is the collection of these peaks at the frequencies that appear in this resolution of the function.

Image (Animated gif): The Fourier transform takes an input function f (in red) in the “time domain” and converts it into a new function f-hat (in blue) in the “frequency domain”. In other words, the original function can be thought of as being “amplitude given time”, and the Fourier transform of the function is “amplitude given frequency”.
Shown here, a simple 6-component approximation of the square wave is decomposed (exactly, for simplicity) into 6 sine waves. These component frequencies show as very sharp peaks in the frequency domain of the function, shown as the blue graph. In practice, these peaks are never that sharp. That would require infinite precision.

Image (gif animation) by Lucas Vieira Barbosa - http://1ucasvb.tumblr.com.

See more: Replicate the Fourier transform time-frequency domains correspondence illustration using TikZ (at here).

spring-of-mathematics:

Mathematical chess problems: Knight’s tour & Longest uncrossed knight’s path.
A knight’s tour is a sequence of moves of a knight on a chessboard such that the knight visits every square only once. If the knight ends on a square that is one knight’s move from the beginning square (so that it could tour the board again immediately, following the same path), the tour is closed, otherwise it is open. The knight’s tour problem is the mathematical problem of finding a knight’s tour. Variations of the knight’s tour problem involve chessboards of different sizes than the usual 8 × 8, as well as irregular (non-rectangular) boards.
The longest uncrossed (or nonintersecting) knight’s path is a mathematical problem involving a knight on the standard 8×8 chessboard or, more generally, on a square n×n board. The problem is to find the longest path the knight can take on the given board, such that the path does not intersect itself. A further distinction can be made between a closed path, which ends on the same field as where it begins, and an open path, which ends on a different field from where it begins.
See more at: Knight’s tour (Figure 1-2) & Longest uncrossed knight’s path (Figure 4-5) on Wikipedia - Knight Graph on Math Wolfram.
Figure 3: Chess and goose game board at The Metropolitan Museum of Art.
Zoom Info
spring-of-mathematics:

Mathematical chess problems: Knight’s tour & Longest uncrossed knight’s path.
A knight’s tour is a sequence of moves of a knight on a chessboard such that the knight visits every square only once. If the knight ends on a square that is one knight’s move from the beginning square (so that it could tour the board again immediately, following the same path), the tour is closed, otherwise it is open. The knight’s tour problem is the mathematical problem of finding a knight’s tour. Variations of the knight’s tour problem involve chessboards of different sizes than the usual 8 × 8, as well as irregular (non-rectangular) boards.
The longest uncrossed (or nonintersecting) knight’s path is a mathematical problem involving a knight on the standard 8×8 chessboard or, more generally, on a square n×n board. The problem is to find the longest path the knight can take on the given board, such that the path does not intersect itself. A further distinction can be made between a closed path, which ends on the same field as where it begins, and an open path, which ends on a different field from where it begins.
See more at: Knight’s tour (Figure 1-2) & Longest uncrossed knight’s path (Figure 4-5) on Wikipedia - Knight Graph on Math Wolfram.
Figure 3: Chess and goose game board at The Metropolitan Museum of Art.
Zoom Info
spring-of-mathematics:

Mathematical chess problems: Knight’s tour & Longest uncrossed knight’s path.
A knight’s tour is a sequence of moves of a knight on a chessboard such that the knight visits every square only once. If the knight ends on a square that is one knight’s move from the beginning square (so that it could tour the board again immediately, following the same path), the tour is closed, otherwise it is open. The knight’s tour problem is the mathematical problem of finding a knight’s tour. Variations of the knight’s tour problem involve chessboards of different sizes than the usual 8 × 8, as well as irregular (non-rectangular) boards.
The longest uncrossed (or nonintersecting) knight’s path is a mathematical problem involving a knight on the standard 8×8 chessboard or, more generally, on a square n×n board. The problem is to find the longest path the knight can take on the given board, such that the path does not intersect itself. A further distinction can be made between a closed path, which ends on the same field as where it begins, and an open path, which ends on a different field from where it begins.
See more at: Knight’s tour (Figure 1-2) & Longest uncrossed knight’s path (Figure 4-5) on Wikipedia - Knight Graph on Math Wolfram.
Figure 3: Chess and goose game board at The Metropolitan Museum of Art.
Zoom Info
spring-of-mathematics:

Mathematical chess problems: Knight’s tour & Longest uncrossed knight’s path.
A knight’s tour is a sequence of moves of a knight on a chessboard such that the knight visits every square only once. If the knight ends on a square that is one knight’s move from the beginning square (so that it could tour the board again immediately, following the same path), the tour is closed, otherwise it is open. The knight’s tour problem is the mathematical problem of finding a knight’s tour. Variations of the knight’s tour problem involve chessboards of different sizes than the usual 8 × 8, as well as irregular (non-rectangular) boards.
The longest uncrossed (or nonintersecting) knight’s path is a mathematical problem involving a knight on the standard 8×8 chessboard or, more generally, on a square n×n board. The problem is to find the longest path the knight can take on the given board, such that the path does not intersect itself. A further distinction can be made between a closed path, which ends on the same field as where it begins, and an open path, which ends on a different field from where it begins.
See more at: Knight’s tour (Figure 1-2) & Longest uncrossed knight’s path (Figure 4-5) on Wikipedia - Knight Graph on Math Wolfram.
Figure 3: Chess and goose game board at The Metropolitan Museum of Art.
Zoom Info
spring-of-mathematics:

Mathematical chess problems: Knight’s tour & Longest uncrossed knight’s path.
A knight’s tour is a sequence of moves of a knight on a chessboard such that the knight visits every square only once. If the knight ends on a square that is one knight’s move from the beginning square (so that it could tour the board again immediately, following the same path), the tour is closed, otherwise it is open. The knight’s tour problem is the mathematical problem of finding a knight’s tour. Variations of the knight’s tour problem involve chessboards of different sizes than the usual 8 × 8, as well as irregular (non-rectangular) boards.
The longest uncrossed (or nonintersecting) knight’s path is a mathematical problem involving a knight on the standard 8×8 chessboard or, more generally, on a square n×n board. The problem is to find the longest path the knight can take on the given board, such that the path does not intersect itself. A further distinction can be made between a closed path, which ends on the same field as where it begins, and an open path, which ends on a different field from where it begins.
See more at: Knight’s tour (Figure 1-2) & Longest uncrossed knight’s path (Figure 4-5) on Wikipedia - Knight Graph on Math Wolfram.
Figure 3: Chess and goose game board at The Metropolitan Museum of Art.
Zoom Info

spring-of-mathematics:

Mathematical chess problems: Knight’s tour & Longest uncrossed knight’s path.

  • A knight’s tour is a sequence of moves of a knight on a chessboard such that the knight visits every square only once. If the knight ends on a square that is one knight’s move from the beginning square (so that it could tour the board again immediately, following the same path), the tour is closed, otherwise it is open. The knight’s tour problem is the mathematical problem of finding a knight’s tour. Variations of the knight’s tour problem involve chessboards of different sizes than the usual 8 × 8, as well as irregular (non-rectangular) boards.
  • The longest uncrossed (or nonintersecting) knight’s path is a mathematical problem involving a knight on the standard 8×8 chessboard or, more generally, on a square n×n board. The problem is to find the longest path the knight can take on the given board, such that the path does not intersect itself. A further distinction can be made between a closed path, which ends on the same field as where it begins, and an open path, which ends on a different field from where it begins.

See more at: Knight’s tour (Figure 1-2) & Longest uncrossed knight’s path (Figure 4-5) on Wikipedia - Knight Graph on Math Wolfram.

Figure 3: Chess and goose game board at The Metropolitan Museum of Art.