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Can you reverse the teapot spin with your mind?

Can you reverse the teapot spin with your mind?

6,833,669 次观看

Can you reverse the chair's rotation with your mind?

Can you reverse the chair's rotation with your mind?

4,026,660 次观看

Can you reverse the duck spin with your mind?

Can you reverse the duck spin with your mind?

131,460 次观看

Can you reverse the human spin with your mind?

Can you reverse the human spin with your mind?

113,440 次观看

Euler's formula states that e^(iθ) = cos θ + i sin θ. As the angle θ increases, e^(iθ) traces the unit circle in the complex plane. Plotted against θ, that rotation becomes a helix. The real part of the helix projects as the cosine wave Re(z) = cos θ. The imaginary part projects as the sine wave Im(z) = sin θ.

Euler's formula states that e^(iθ) = cos θ + i sin θ. As the angle θ increases, e^(iθ) traces the unit circle in the complex plane. Plotted against θ, that rotation becomes a helix. The real part of the helix projects as the cosine wave Re(z) = cos θ. The imaginary part projects as the sine wave Im(z) = sin θ.

55,150 次观看

Can you reverse the spin of this elephant with your mind?

Can you reverse the spin of this elephant with your mind?

51,413 次观看

A closed outline shaped like the Greek letter pi can be rebuilt as the path of a point carried by a chain of rotating circles, each circle matching one frequency component through its radius and phase, so that successive circles refine the match to the original curve.

A closed outline shaped like the Greek letter pi can be rebuilt as the path of a point carried by a chain of rotating circles, each circle matching one frequency component through its radius and phase, so that successive circles refine the match to the original curve.

12,050 次观看

A pendulum bob midway through its downward swing shows gravitational potential energy converting into kinetic energy, visible as a medium-height blue bar for remaining potential energy, a short orange bar for kinetic energy, and a taller cyan bar of constant height representing the conserved total mechanical energy.

A pendulum bob midway through its downward swing shows gravitational potential energy converting into kinetic energy, visible as a medium-height blue bar for remaining potential energy, a short orange bar for kinetic energy, and a taller cyan bar of constant height representing the conserved total mechanical energy.

12,804 次观看

When a drone applies a force at an angle to a cart resting on a level surface the force produces a horizontal pull that moves the cart forward and a vertical lift that lightens the load on the wheels, illustrating how vectors separate into independent effects along two perpendicular axes.

When a drone applies a force at an angle to a cart resting on a level surface the force produces a horizontal pull that moves the cart forward and a vertical lift that lightens the load on the wheels, illustrating how vectors separate into independent effects along two perpendicular axes.

13,022 次观看

Quadric Morph continuously deforms a sphere through a family of quadric surfaces into a hyperboloid of two sheets, showing the transition from a closed connected shape to two separate open bowls and thereby illustrating the classification of these surfaces in analytic geometry.

Quadric Morph continuously deforms a sphere through a family of quadric surfaces into a hyperboloid of two sheets, showing the transition from a closed connected shape to two separate open bowls and thereby illustrating the classification of these surfaces in analytic geometry.

13,372 次观看

3D rotation matrices about coordinate axes. Rₓ(θ) fixes x and rotates yz-plane [[1 0 0] [0 cosθ −sinθ] [0 sinθ cosθ]]. Rᵧ(θ) fixes y and rotates xz-plane [[cosθ 0 sinθ] [0 1 0] [−sinθ 0 cosθ]]. Rz(θ) fixes z and rotates xy-plane [[cosθ −sinθ 0] [sinθ cosθ 0] [0 0 1]].

3D rotation matrices about coordinate axes. Rₓ(θ) fixes x and rotates yz-plane [[1 0 0] [0 cosθ −sinθ] [0 sinθ cosθ]]. Rᵧ(θ) fixes y and rotates xz-plane [[cosθ 0 sinθ] [0 1 0] [−sinθ 0 cosθ]]. Rz(θ) fixes z and rotates xy-plane [[cosθ −sinθ 0] [sinθ cosθ 0] [0 0 1]].

16,193 次观看

Are you able to reverse the spin with your mind?

Are you able to reverse the spin with your mind?

33,135 次观看

Electromagnetic radiation: oscillating electric and magnetic fields propagate outward from a source at the speed of light. The fields E and B remain mutually perpendicular and orthogonal to the radial unit vector r̂. Intensity I of the radiation decreases with the inverse square of the distance, I ∝ r⁻².

Electromagnetic radiation: oscillating electric and magnetic fields propagate outward from a source at the speed of light. The fields E and B remain mutually perpendicular and orthogonal to the radial unit vector r̂. Intensity I of the radiation decreases with the inverse square of the distance, I ∝ r⁻².

13,498 次观看

$A mathematical construction in which one strip completes exactly ten turns. The polar angle of the generating curve is controlled by the exponential φ(θ) = (π/2) e^{-θ/8π}. Successive windings therefore nest inside one another, forming a compact layered rose with an exponentially tapering profile.$

A mathematical construction in which one strip completes exactly ten turns. The polar angle of the generating curve is controlled by the exponential φ(θ) = (π/2) e^{-θ/8π}. Successive windings therefore nest inside one another, forming a compact layered rose with an exponentially tapering profile.

13,530 次观看

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All six trigonometric functions appear as distinct line-segment lengths on a single unit circle.

All six trigonometric functions appear as distinct line-segment lengths on a single unit circle.

28,810 次观看 • 14 天前

A current-carrying coil placed between the north and south poles of a magnet experiences forces on opposite sides that produce a continuous turning effect, while the battery supplies the current and the commutator keeps the rotation unidirectional as both the terminal voltage and the torque change with the coil’s angular position.

A current-carrying coil placed between the north and south poles of a magnet experiences forces on opposite sides that produce a continuous turning effect, while the battery supplies the current and the commutator keeps the rotation unidirectional as both the terminal voltage and the torque change with the coil’s angular position.

13,326 次观看 • 7 天前

In three-dimensional space the determinant of a matrix whose columns are three vectors equals the signed volume of the parallelepiped those vectors span; changing the middle column so the base parallelogram doubles in area while height stays fixed therefore yields a volume of two cubic units.

In three-dimensional space the determinant of a matrix whose columns are three vectors equals the signed volume of the parallelepiped those vectors span; changing the middle column so the base parallelogram doubles in area while height stays fixed therefore yields a volume of two cubic units.

16,450 次观看 • 9 天前

$The Gaussian integral evaluates to √π: ∫_{−∞}^{∞} e^{−x²} dx = √π. The shaded region under y = e^{−x²} has this exact area. Letting I denote the integral, I² equals the volume of the surface z = e^{−(x² + y²)}, a rotationally symmetric hill whose volume is π when computed in polar coordinates.$

The Gaussian integral evaluates to √π: ∫_{−∞}^{∞} e^{−x²} dx = √π. The shaded region under y = e^{−x²} has this exact area. Letting I denote the integral, I² equals the volume of the surface z = e^{−(x² + y²)}, a rotationally symmetric hill whose volume is π when computed in polar coordinates.

28,660 次观看 • 18 天前

Learn this graph through the dance of the blue stick figure, whose raised arms form a wide V on the coordinate grid while red lines extend the pose to the corners.

Learn this graph through the dance of the blue stick figure, whose raised arms form a wide V on the coordinate grid while red lines extend the pose to the corners.

20,496 次观看 • 14 天前

Group theory lives inside the cube. Inverse: R R′ = I. A right face turn followed by its inverse returns the identity. Order: R⁴ = I. Four quarter turns of one face restore the original position. Commutator: A B A⁻¹ B⁻¹ ≠ I. Adjacent moves generally fail to commute. Every cube move obeys these rules.

Group theory lives inside the cube. Inverse: R R′ = I. A right face turn followed by its inverse returns the identity. Order: R⁴ = I. Four quarter turns of one face restore the original position. Commutator: A B A⁻¹ B⁻¹ ≠ I. Adjacent moves generally fail to commute. Every cube move obeys these rules.

17,237 次观看 • 18 天前

A point traces each closed mathematical curve at constant speed. Its vertical coordinate versus elapsed time produces a unique periodic waveform. 4-petal rose r = a cos(2θ). Astroid hypocycloid x = a cos³θ, y = a sin³θ. Squircle |x|⁴ + |y|⁴ = a⁴. 4-cusped epicycloid. Heart curve (x² + y² - 1)³ = x² y³.

A point traces each closed mathematical curve at constant speed. Its vertical coordinate versus elapsed time produces a unique periodic waveform. 4-petal rose r = a cos(2θ). Astroid hypocycloid x = a cos³θ, y = a sin³θ. Squircle |x|⁴ + |y|⁴ = a⁴. 4-cusped epicycloid. Heart curve (x² + y² - 1)³ = x² y³.

10,607 次观看 • 15 天前

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