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22/03/2024 *
A Bicorne is a historical hat worn by American and European army and naval officers. It is shaped like the graph produced by this implicit equation!

Check out the link below to see it graphed!

https://mathonthecloud.com/realfunctions/implicitmulti?&function0=y%5E2*(a%5E2-x%5E2)-(x%5E2%2B2*a*y-a%5E2)%5E2&colors=10%2C9%2C8%2C7%2C6%2C5%2C4%2C3%2C2%2C1&xmin=-10&xmax=10&ymin=-5&ymax=15

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19/03/2024 *
A rather simple polar equation, r=sin(θ)cos(θ)^2+sin(2θ), creates this simple and iconic butterfly when graphed! 🦋🦋🦋

Check out the link below to see it graphed on mathonthecloud.com!

https://mathonthecloud.com/realfunctions/polar?&function0=sin(a)*cos(a)%5E2%2Bsin(2*a)&amin=0&amax=7

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14/03/2024 *
The polar function r=1-3cos(5θ)+2sin(5θ)^4 takes the shape of a cherry blossom when graphed! 🌸🌸🌸

Check out the link below to see it graphed!

https://mathonthecloud.com/realfunctions/polar?&function0=1-3*cos(5*a)%2B2*sin(5*a)%5E4&amin=0&amax=7

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07/03/2024 *
This seashell-shaped curve has the implicit equation 4*(x^2+y^2-a^2)^3-27*a^4*y=0. The parameter 'a' determines the size of the graph. 🐚

Check out the link below to see it graphed!

https://mathonthecloud.com/realfunctions/implicitmulti?&function0=4*(x%5E2%2By%5E2-a%5E2)%5E3-27*a%5E4*y&colors=1%2C2%2C3%2C4%2C5%2C6%2C7%2C8%2C9%2C10&xmin=-15&xmax=15&ymin=-10&ymax=20

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05/03/2024 *
The implicit equation a^4*(1.5*y)^2-(a^2-x^2)^3=0 produces this interesting graph. Some say they look like puckered lips, but I think they are the outlines of a stereotypical UFO!

Check out the link below!

https://mathonthecloud.com/realfunctions/implicitmulti?&function0=a%5E4*(1.5*y)%5E2-(a%5E2-x%5E2)%5E3&colors=1%2C2%2C3%2C4%2C5%2C6%2C7%2C8%2C9%2C10&xmin=-10&xmax=10&ymin=-10&ymax=10

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03/03/2024 *
Fermat's spiral is a parabolic spiral with the implicit equation y*sin((x^2+y^2)/a^2)=x*cos((x^2+y^2)/a^2). It also has a polar equation of r^2=a^2*θ. It is named after the famous French mathematician who studied this curve in the 1600s!

Check out the link below to see it graphed!

https://mathonthecloud.com/realfunctions/implicit -> https://t.co/27NBPGaAIA

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27/02/2024 *
This ovoid can be graphed with the implicit function (x^2+y^2)^2-a*x^3=0, or the polar function r=a*cos(θ)^3, with 'a' as a parameter that determines its size. It has a distinctly accurate egg shape! https://mathonthecloud.com/realfunctions/implicitmulti?&function0=(x%5E2%2By%5E2)%5E2-a*x%5E3&colors=1%2C2%2C3%2C4%2C5%2C6%2C7%2C8%2C9%2C10&xmin=-1&xmax=11&ymin=-6&ymax=6

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26/02/2024 *
This teardrop-shaped curve is called the Piriform Curve. It is made with the function family 8^4*y^2-x^3*a^2*(2*8-x)=0, where 'a' is an independent parameter.

Click on the link to see it graphed! https://mathonthecloud.com/realfunctions/implicitmulti?&function0=8%5E4*y%5E2-x%5E3*a%5E2*(2*8-x)&colors=1%2C2%2C3%2C4%2C5%2C6%2C7%2C8%2C9&xmin=-6&xmax=24&ymin=-15&ymax=15

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25/02/2024 *
The Klein quartic, also known as 'a funny curve,' has the implicit function of x^3*y+y^3*z+z^3*x=0. It is graphed here as a function family with 'z' as an independent parameter. Click on the link to see it graphed!

https://mathonthecloud.com/realfunctions/implicitmulti?&function0=x%5E3*y%2By%5E3*a%2Ba%5E3*x&colors=-7%2C-6%2C-5%2C-4%2C-3%2C-2%2C-1%2C0%2C1%2C2%2C3%2C4%2C5%2C6%2C7&xmin=-8&xmax=8&ymin=-8&ymax=8

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24/02/2024 *
A complex function w=F(z) will map points, lines/graphs, or other objects in one complex plane(z) to another(w). Here, a rectangular grid is transformed through the function w=1/z. Try it out with your own functions! https://mathonthecloud.com/complexfunctions/complexmap?&function0=1%2Fz&Xc=20&Yc=0&Dxy=10

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22/02/2024 *
All explicit functions can be rewritten as implicit functions, but not always the other way around! The explicit function y=sin(2*(x-3*a)) has an implicit form of y-sin(2*(x-3*a))=0.

Click on the link below to graph your own functions!

https://mathonthecloud.com/realfunctions/implicitmulti -> https://t.co/JQHTGMktK2

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21/02/2024 *
The polar function r=1-a*sin(b*θ) produces a two layered, flower-shaped curve. Altering the a and b parameters changes the size and number of petals! You can try it out by visiting the following link! https://mathonthecloud.com/realfunctions/polar?&function0=1-4*sin(8*a)&amin=0&amax=100

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16/02/2024 *
This diamond (or rhombus) curve is graphed with the implicit equation of |x|+|y|-a=0, where parameter "a" determines its size. It's a special case of the superellipse, an ellipse with symmetry about the x and y axes. https://mathonthecloud.com/realfunctions/implicitmulti?&function0=abs(x)%2Babs(y)-a&colors=1%2C2%2C3%2C4%2C5%2C6%2C7%2C8%2C9%2C10%2C11&xmin=-12&xmax=12&ymin=-12&ymax=12

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01/02/2024 *
A slinky-like function! The x and y coordinates of a parametric function graph are dependent on an independent variable, t, rather than each other. https://mathonthecloud.com/realfunctions/parametric -> https://t.co/OxS3pOgQtG

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28/01/2024 *
🌀🌀 The Golden Spiral is a logarithmic spiral with the growth factor of φ(phi), the golden ratio, which is approximately 1.618. Logarithmic spirals can often be found in nature, such as the shells of mollusks, in the form of hurricanes, and the shape of the Milky Way galaxy. https://mathonthecloud.com/realfunctions/polar -> https://t.co/rvqwddJ4AJ

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25/01/2024 *
✠✠ The Maltese Cross is an ancient symbol that dates back to 16th century Europe. It represents bravery and its imagery often serves as service medals all over the world. It can be represented by implicit and polar equations! Here, the parameter 'a' controls the size of the cross. https://mathonthecloud.com/realfunctions/implicitmulti?&function0=a%2F2*x*y*(x%5E2-y%5E2)-x%5E2-y%5E2&colors=1%2C1.5%2C2%2C3%2C4%2C5%2C7%2C10%2C12&xmin=-5&xmax=5&ymin=-5&ymax=5

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21/01/2024 *
🌼🌼The Dürer Folium is a rose curve with the polar functions r=a*sin(θ/2) or r=a*cos(θ/2). (Though unequal, both of these equations will result in the same graph!) It also has a more complicated implicit equation! https://mathonthecloud.com/realfunctions/implicitmulti?&function0=a%5E4*y%5E2%2B4*(x%5E2%2By%5E2)%5E3-4*a%5E2*(x%5E2%2By%5E2)%5E2&colors=3%2C4%2C5%2C6%2C7%2C8%2C9%2C10&xmin=-11&xmax=11&ymin=-11&ymax=11

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18/01/2024 *
🏋️ This Dumbbell Curve is a sextic(6th-degree) curve with the equation x^4*(a^2-x^2)-(0.8*a)^4*y^2=0. Here, the parameter 'a' controls the overall size and the 'weight' ends of the dumbbell. https://mathonthecloud.com/realfunctions/implicitmulti?&function0=x%5E4*(a%5E2-x%5E2)-(0.8*a)%5E4*y%5E2&colors=2%2C4%2C5%2C6%2C7%2C8%2C9%2C10%2C11%2C12&xmin=-12.5&xmax=12.5&ymin=-12.5&ymax=12.5

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16/01/2024 *
An excitable ellipse! As the parameter 'a' increases in the implicit function (x-sin(a*x))^2+(5*y^2)/7-a=0, the more chaotic the ellipse becomes! https://mathonthecloud.com/realfunctions/implicitmulti => https://t.co/k3qylhCxyP

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13/01/2024 *
The Devil's Curve is so named for the central lemniscate (figure-8) shape that resembles a diabolo. Of course, there's nothing evil about a math equation! 😈😈😈 Here it is graphed with a single changing parameter. https://mathonthecloud.com/realfunctions/implicitmulti?&function0=y%5E4-(a%2F3)%5E2*y%5E2-x%5E4%2B(a%2F2)%5E2*x%5E2&colors=1%2C2%2C3%2C4%2C5%2C6%2C7%2C8%2C9%2C10&xmin=-6&xmax=6&ymin=-6&ymax=6

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10/01/2024 *
Very intricate designs are often made with clever math! Here is a rather complicated polar function utilizing rounding that results in a pinwheel-like graph! ✵✵✵ Check it out graphed on http://mathonthecloud.com! -> https://t.co/L4FIsqqyQX

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08/01/2024 *
A family of implicit functions that graphs into clovers 🍀🍀! In (x^2+y^2)^3-4*x^2*y^2*(a^2+1)=a/10000, 'a' parameter determines the size of each clover. The right side value controls the distinctness of the pedals. https://mathonthecloud.com/realfunctions/implicitmulti?&function0=(x%5E2%2By%5E2)%5E3-4*x%5E2*y%5E2*(a%5E2%2B1)-a%2F10000&colors=1%2C2%2C3%2C4%2C5%2C6%2C7%2C8%2C9%2C10&xmin=-10&xmax=10&ymin=-10&ymax=10