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).
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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
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
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

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).

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
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
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

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.

curiosamathematica:

Remember my post about Sangakus?
Some time ago our local math association PRIME organized Soiree Sangaku, where the participants could investigate 30 of these Japanese puzzles. Try to solve these for yourselves!
I call the first one “one rule to Ring them all”. The puzzle is to find (and prove) the relation between the rings. In the second and third one, try find a relation between the radii of the orange circles. Warning: the latter aren’t easy at all… Good luck!
If you like these problems but got stuck at some point,don’t hesitate to contact me ;)
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curiosamathematica:

Remember my post about Sangakus?
Some time ago our local math association PRIME organized Soiree Sangaku, where the participants could investigate 30 of these Japanese puzzles. Try to solve these for yourselves!
I call the first one “one rule to Ring them all”. The puzzle is to find (and prove) the relation between the rings. In the second and third one, try find a relation between the radii of the orange circles. Warning: the latter aren’t easy at all… Good luck!
If you like these problems but got stuck at some point,don’t hesitate to contact me ;)
Zoom Info
curiosamathematica:

Remember my post about Sangakus?
Some time ago our local math association PRIME organized Soiree Sangaku, where the participants could investigate 30 of these Japanese puzzles. Try to solve these for yourselves!
I call the first one “one rule to Ring them all”. The puzzle is to find (and prove) the relation between the rings. In the second and third one, try find a relation between the radii of the orange circles. Warning: the latter aren’t easy at all… Good luck!
If you like these problems but got stuck at some point,don’t hesitate to contact me ;)
Zoom Info

curiosamathematica:

Remember my post about Sangakus?

Some time ago our local math association PRIME organized Soiree Sangaku, where the participants could investigate 30 of these Japanese puzzles. Try to solve these for yourselves!

I call the first one “one rule to Ring them all”. The puzzle is to find (and prove) the relation between the rings. In the second and third one, try find a relation between the radii of the orange circles. Warning: the latter aren’t easy at all… Good luck!

If you like these problems but got stuck at some point,
don’t hesitate to contact me ;)

Which of the marked points belongs to the interior of the polygon? (Fig.3)
In topology, a Jordan curve is a non-self-intersecting continuous loop in the plane, and another name for a Jordan curve is a simple closed curve. The Jordan curve theorem asserts that every Jordan curve divides the plane into an “interior” region bounded by the curve and an “exterior” region containing all of the nearby and far away exterior points, so that any continuous path connecting a point of one region to a point of the other intersects with that loop somewhere. While the statement of this theorem seems to be intuitively obvious, it takes quite a bit of ingenuity to prove it by elementary means. More transparent proofs rely on the mathematical machinery of algebraic topology, and these lead to generalizations to higher-dimensional spaces. (Source: Jordan curve theorem on Wiki).
Image: [1] - [2]- [3]
See more at:  http://en.wikipedia.org/wiki/Jordan_curve_theorem & http://mathworld.wolfram.com/JordanCurve.html.
Zoom Info
Which of the marked points belongs to the interior of the polygon? (Fig.3)
In topology, a Jordan curve is a non-self-intersecting continuous loop in the plane, and another name for a Jordan curve is a simple closed curve. The Jordan curve theorem asserts that every Jordan curve divides the plane into an “interior” region bounded by the curve and an “exterior” region containing all of the nearby and far away exterior points, so that any continuous path connecting a point of one region to a point of the other intersects with that loop somewhere. While the statement of this theorem seems to be intuitively obvious, it takes quite a bit of ingenuity to prove it by elementary means. More transparent proofs rely on the mathematical machinery of algebraic topology, and these lead to generalizations to higher-dimensional spaces. (Source: Jordan curve theorem on Wiki).
Image: [1] - [2]- [3]
See more at:  http://en.wikipedia.org/wiki/Jordan_curve_theorem & http://mathworld.wolfram.com/JordanCurve.html.
Zoom Info
Which of the marked points belongs to the interior of the polygon? (Fig.3)
In topology, a Jordan curve is a non-self-intersecting continuous loop in the plane, and another name for a Jordan curve is a simple closed curve. The Jordan curve theorem asserts that every Jordan curve divides the plane into an “interior” region bounded by the curve and an “exterior” region containing all of the nearby and far away exterior points, so that any continuous path connecting a point of one region to a point of the other intersects with that loop somewhere. While the statement of this theorem seems to be intuitively obvious, it takes quite a bit of ingenuity to prove it by elementary means. More transparent proofs rely on the mathematical machinery of algebraic topology, and these lead to generalizations to higher-dimensional spaces. (Source: Jordan curve theorem on Wiki).
Image: [1] - [2]- [3]
See more at:  http://en.wikipedia.org/wiki/Jordan_curve_theorem & http://mathworld.wolfram.com/JordanCurve.html.
Zoom Info

Which of the marked points belongs to the interior of the polygon? (Fig.3)

In topology, a Jordan curve is a non-self-intersecting continuous loop in the plane, and another name for a Jordan curve is a simple closed curve. The Jordan curve theorem asserts that every Jordan curve divides the plane into an “interior” region bounded by the curve and an “exterior” region containing all of the nearby and far away exterior points, so that any continuous path connecting a point of one region to a point of the other intersects with that loop somewhere. While the statement of this theorem seems to be intuitively obvious, it takes quite a bit of ingenuity to prove it by elementary means. More transparent proofs rely on the mathematical machinery of algebraic topology, and these lead to generalizations to higher-dimensional spaces. (Source: Jordan curve theorem on Wiki).

Image: [1] - [2][3]

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

Proof: (1+2+3+….+n)^2 = 1^3+2^3+3^3+….+n^3.
Explains this Image:
(1+2+3+4+5+6+7+8)^2 = 1^3+2^3+3^3+4^3+5^3+6^3+7^3+8^3
S(square) = (1+2+3+4+5+6+7+8)x(1+2+3+4+5+6+7+8) = (1+2+3+4+5+6+7+8)^2
Also, S(square) = SUM of small squares = 1x1^2 + 2x(2^2) + 3x(3^2) + 4x(4^2)+…….+8x(8^2) = 1^3 + 2^3 + 3^3 + 4^3 + 5^3 + 6^3 + 7^3 + 8^3
(nx(n^2) mean n squares have S = n^2)
In there, pink square and white square are compensate for each other.
See more: 3D geometry proof posted by Hyrodium’s Graphical MathLand &TwoCubes.
Image: Carre de la somme des 8 premiers entiers on Wikipedia.

Proof: (1+2+3+….+n)^2 = 1^3+2^3+3^3+….+n^3.

Explains this Image:

(1+2+3+4+5+6+7+8)^2 = 1^3+2^3+3^3+4^3+5^3+6^3+7^3+8^3

S(square) = (1+2+3+4+5+6+7+8)x(1+2+3+4+5+6+7+8) = (1+2+3+4+5+6+7+8)^2

Also, S(square) = SUM of small squares = 1x1^2 + 2x(2^2) + 3x(3^2) + 4x(4^2)+…….+8x(8^2) = 1^3 + 2^3 + 3^3 + 4^3 + 5^3 + 6^3 + 7^3 + 8^3

(nx(n^2) mean n squares have S = n^2)

In there, pink square and white square are compensate for each other.

See more: 3D geometry proof posted by Hyrodium’s Graphical MathLand &TwoCubes.

Image: Carre de la somme des 8 premiers entiers on Wikipedia.

Bézier Clock
About Bézier Clock: This is a program with  Processing.js  (see code); ; it just linearly interpolates five Bézier control points between pre-defined states. Bézier curves are made up of various ‘node’ points that the curve passes through, and ‘control’ points for each node that affect the local curvature. By adjust the locations and distance of the control points from the node points, you can create complex curves with very few points. What this program does is calculate straight-line paths between each point’s location on the current and next shown digit, and draws a Bézier curve through the interpolated points.
Source: Bézier Clock by Jack Frigaard - The interactive version, introductions can be found here at http://jackf.net/bezier-clock/.
Image: Shared at http://i.imgur.com/5hFDMn3.gif
BEZIER CURVES:This spline approximation method was developed by the French engineer Pierre Bezier for use in the design of Renault automobile bodies. Bezier splines have a number of properties that make them highly useful and convenient for curve and surface design. They are also easy to implement. For these reasons, Bezier splines are widely available in various CAD system.In general, a Bezier curve section can be fitted to any number of control points. The number of control points to be approximated and their relative position determine the degree of the Bezier polynomial. As with a charactering matrix, or with blending functions. For general Bezier curves, the blending-function specification is the most convenient.  (See more at Bézier curves).As a rule, a Bezier curve is a polynomial of degree one less than the number of control points used: Three points generate a parabola, four points a cube curve, and so forth. Figure 1 demonstrates the appearance of some Bezier curves for various selections of control points in the xy plane (z = 0). With certain control-point placements, however, we obtain degenerate Bezier polynomials. For example, a Bezier “curve” that is a single point.[Source]Bezier curves are commonly found in painting and drawing packages, as well as CAD system, since they are easy to implement and they are reasonably powerful in curve design. Efficient methods for determining coordinate positions along a Bezier curve can be set up using recursive calculations. For example, successive binomial coefficients can be calculates as.
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Bézier Clock
About Bézier Clock: This is a program with  Processing.js  (see code); ; it just linearly interpolates five Bézier control points between pre-defined states. Bézier curves are made up of various ‘node’ points that the curve passes through, and ‘control’ points for each node that affect the local curvature. By adjust the locations and distance of the control points from the node points, you can create complex curves with very few points. What this program does is calculate straight-line paths between each point’s location on the current and next shown digit, and draws a Bézier curve through the interpolated points.
Source: Bézier Clock by Jack Frigaard - The interactive version, introductions can be found here at http://jackf.net/bezier-clock/.
Image: Shared at http://i.imgur.com/5hFDMn3.gif
BEZIER CURVES:This spline approximation method was developed by the French engineer Pierre Bezier for use in the design of Renault automobile bodies. Bezier splines have a number of properties that make them highly useful and convenient for curve and surface design. They are also easy to implement. For these reasons, Bezier splines are widely available in various CAD system.In general, a Bezier curve section can be fitted to any number of control points. The number of control points to be approximated and their relative position determine the degree of the Bezier polynomial. As with a charactering matrix, or with blending functions. For general Bezier curves, the blending-function specification is the most convenient.  (See more at Bézier curves).As a rule, a Bezier curve is a polynomial of degree one less than the number of control points used: Three points generate a parabola, four points a cube curve, and so forth. Figure 1 demonstrates the appearance of some Bezier curves for various selections of control points in the xy plane (z = 0). With certain control-point placements, however, we obtain degenerate Bezier polynomials. For example, a Bezier “curve” that is a single point.[Source]Bezier curves are commonly found in painting and drawing packages, as well as CAD system, since they are easy to implement and they are reasonably powerful in curve design. Efficient methods for determining coordinate positions along a Bezier curve can be set up using recursive calculations. For example, successive binomial coefficients can be calculates as.
Zoom Info
Bézier Clock
About Bézier Clock: This is a program with  Processing.js  (see code); ; it just linearly interpolates five Bézier control points between pre-defined states. Bézier curves are made up of various ‘node’ points that the curve passes through, and ‘control’ points for each node that affect the local curvature. By adjust the locations and distance of the control points from the node points, you can create complex curves with very few points. What this program does is calculate straight-line paths between each point’s location on the current and next shown digit, and draws a Bézier curve through the interpolated points.
Source: Bézier Clock by Jack Frigaard - The interactive version, introductions can be found here at http://jackf.net/bezier-clock/.
Image: Shared at http://i.imgur.com/5hFDMn3.gif
BEZIER CURVES:This spline approximation method was developed by the French engineer Pierre Bezier for use in the design of Renault automobile bodies. Bezier splines have a number of properties that make them highly useful and convenient for curve and surface design. They are also easy to implement. For these reasons, Bezier splines are widely available in various CAD system.In general, a Bezier curve section can be fitted to any number of control points. The number of control points to be approximated and their relative position determine the degree of the Bezier polynomial. As with a charactering matrix, or with blending functions. For general Bezier curves, the blending-function specification is the most convenient.  (See more at Bézier curves).As a rule, a Bezier curve is a polynomial of degree one less than the number of control points used: Three points generate a parabola, four points a cube curve, and so forth. Figure 1 demonstrates the appearance of some Bezier curves for various selections of control points in the xy plane (z = 0). With certain control-point placements, however, we obtain degenerate Bezier polynomials. For example, a Bezier “curve” that is a single point.[Source]Bezier curves are commonly found in painting and drawing packages, as well as CAD system, since they are easy to implement and they are reasonably powerful in curve design. Efficient methods for determining coordinate positions along a Bezier curve can be set up using recursive calculations. For example, successive binomial coefficients can be calculates as.
Zoom Info

Bézier Clock

About Bézier Clock: This is a program with  Processing.js  (see code); ; it just linearly interpolates five Bézier control points between pre-defined states. Bézier curves are made up of various ‘node’ points that the curve passes through, and ‘control’ points for each node that affect the local curvature. By adjust the locations and distance of the control points from the node points, you can create complex curves with very few points. What this program does is calculate straight-line paths between each point’s location on the current and next shown digit, and draws a Bézier curve through the interpolated points.

Source: Bézier Clock by Jack Frigaard - The interactive version, introductions can be found here at http://jackf.net/bezier-clock/.

Image: Shared at http://i.imgur.com/5hFDMn3.gif

BEZIER CURVES:
This spline approximation method was developed by the French engineer Pierre Bezier for use in the design of Renault automobile bodies. Bezier splines have a number of properties that make them highly useful and convenient for curve and surface design. They are also easy to implement. For these reasons, Bezier splines are widely available in various CAD system.
In general, a Bezier curve section can be fitted to any number of control points. The number of control points to be approximated and their relative position determine the degree of the Bezier polynomial. As with a charactering matrix, or with blending functions. For general Bezier curves, the blending-function specification is the most convenient.  (See more at Bézier curves).
As a rule, a Bezier curve is a polynomial of degree one less than the number of control points used: Three points generate a parabola, four points a cube curve, and so forth. Figure 1 demonstrates the appearance of some Bezier curves for various selections of control points in the xy plane (z = 0). With certain control-point placements, however, we obtain degenerate Bezier polynomials. For example, a Bezier “curve” that is a single point.[Source]

Bezier curves are commonly found in painting and drawing packages, as well as CAD system, since they are easy to implement and they are reasonably powerful in curve design. Efficient methods for determining coordinate positions along a Bezier curve can be set up using recursive calculations. For example, successive binomial coefficients can be calculates as.