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**Sumasoltin****Member**- Registered: 2012-04-28
- Posts: 30

I'm a Chinese high school student who love equation and function CRAZILY ! And I've found a lot about this mathematical game.Let's look into!

Absolution,I think it useful but unpleasant,can make any project practicable.As is known to all,the equation "ab=0" is equal to"a=0 or b=0".Similarly,"g(x,y)f(x,y)=0" is equal to "g(x,y)=0 or f(x,y)=0",so now you can use "plus" to combine different plots.And another equation "f²(x,y)+g²(x,y)=0" is equal to "f(x,y)=0 and g(x,y)=0",and in this way you can cut a part of plot.How? There comes the equation "|x-a|+|x-b|-|a-b|=0",whose graph is a part of the plane.Now,let's graph "(x-y)²+(|x-1|+|x+1|-2)²=0".It's a segment.We find |x-1|+|x+1|-2>0,so this equation can be simplified as (x-y)²+|x-1|+|x+1|=2.

This is easy,but not best.Keep patient and you're due to find an equation showing special meanings.Take your aim apart and find equations describe them,and combine them together.

But I dislike absolution,so I found the "fusion".We plus "x=1" and "(x+1/2)²+y²=1",and then we got "x((x+1/2)²+y²)=1".whose plot is the fusion of a circle and straight line.Similarly,many plots can be combine in this way.

There are lots of tips but I can only convey those with my poor English.To show more about this geeks' game I will show you some my original equations.

What's more,have you found some interesting graphs of equation?Share it if it's convenient to you!

*Last edited by Sumasoltin (2012-04-28 22:18:03)*

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**Sumasoltin****Member**- Registered: 2012-04-28
- Posts: 30

Aaaaa......the most beautiful one is larger than 100kb

The function is:

y=abs[x]-abs[x/2+3]-abs[x/2-3]+3(x-14)(x-16)/abs[2(x-14)(x-16)]+3(x-12)(x-18)/abs[(x-12)(x-18)]+(sqrt[4-(abs[x-14]-abs[x-16])^2]+sqrt[36-(abs[x-12]-abs[x-18])^2])/4-3+(18+abs[2x+24]+abs[2x-24]-abs[2x+12]-abs[2x-12]-6(x-14)(x-16)/abs[(x-14)(x-16)]-6(x-12)(x-18)/abs[(x-12)(x-18)]-18(x+14)(x+16)/abs[(x+14)(x+16)]+sqrt[4-(abs[x-14]-abs[x-16])^2]-sqrt[36-(abs[x-12]-abs[x-18])^2])sin[5pi*x]/4

Everyone knows that the heart cruve is a closed cruve so we can't find a functiom with a heart-shaped plot.Really?

y=x^(2/3)+sqrt(4-x^2)sin(5pi*x)

*Last edited by Sumasoltin (2012-05-01 04:41:19)*

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,164

Hi Sumasoltin;

Welcome to the forum.

For your heart curve, what is the interval for x?

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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**Sumasoltin****Member**- Registered: 2012-04-28
- Posts: 30

Hi bobbym;

Thank you.I'm sorry to have forgot to say that.The first one is from x=-25 to 20,and the second is in [-5,5]

I found something interesting I set down before:Nike equation

*Last edited by Sumasoltin (2012-04-29 23:19:01)*

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,544

These images are so cool! How did you derive the heart curve and Nike curve?

Here lies the reader who will never open this book. He is forever dead.

Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,164

Hi Sumasoltin;

What are you using to do the graphing?

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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**Sumasoltin****Member**- Registered: 2012-04-28
- Posts: 30

Hi bobbym;

There are lots of mathematical softwares,like matlb,mathematic,and to graph equations I prefer GrafEq(w w w .xamuel.com/graphs-of-implicit-equations)

*Last edited by Sumasoltin (2012-04-30 00:05:31)*

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,164

y=x^(2/3)+sqrt(4-x^2)sin(5pi*x)

Are you sure this is the right equation?

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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**Sumasoltin****Member**- Registered: 2012-04-28
- Posts: 30

Hi anonimnystefy;

Aaaa......they are created in different ways.Some are using the transformation in which we replace x by x±f(x).

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**Sumasoltin****Member**- Registered: 2012-04-28
- Posts: 30

To bobbym;

Aaaa...what's wrong?

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,164

Nothing yet, I am rechecking, probably the program.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**Sumasoltin****Member**- Registered: 2012-04-28
- Posts: 30

Oh,I got it ! Different softwares define "x^(2/3)" differently.It should be (x^2)^(1/3)

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,544

Still not a heart graph!

Here lies the reader who will never open this book. He is forever dead.

Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment

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**Sumasoltin****Member**- Registered: 2012-04-28
- Posts: 30

WolframAlpha shows as this.

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,544

Oh,yes! Got it now! Thanks for the cool graph! Got more?

Here lies the reader who will never open this book. He is forever dead.

Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,164

Yes, I got that, I did not see the heart at first.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,544

Here's another heart graph that you get by varying the constant with pi*x inside the sine function:

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,164

What is cbrt?

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,544

Cube root function.

Here lies the reader who will never open this book. He is forever dead.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,164

Got MIF's graph to do it.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,544

Good! It is nice. Did you try using sin(k*pi*x) instead of sin(5*pi*x) for arbitrary values of k?

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**Sumasoltin****Member**- Registered: 2012-04-28
- Posts: 30

ρ=1-cosθ , an old equation shows heart cruve (bottom instead I think).But I found "(x²+y²-1)³=x²y³" on WolframAlpha.

"Who found this?",I search it on Google and finally I learned it was Siehe Beutel.But how he found this?

We chosed Archimedes spiral: x²+y²=arc tan²(y/x)

we find that "tan x" is simillar to "x³",so we got

(x²+y²)³=(y/x)²

we find heart isn't a symmetrical shape,so we change "y²" by "y³"

As x becomes larger,the difference between "tan x" and "x³" becomes larger,so we plus "x⁴"

Aha~just a joke ! Maybe following can be turth.

He used a simple ellipse "x²+y²-xy=1".

We transform it to "x²+y²-1=|x|y".

NO ABS ! We get "(x²+y²-1)²=x²y²".

But now we lose the relation of that "x²+y²-1" is the same to "y" in "±" (I can't convey in proper way)

So we try the equation "(a²x²+b²y²-1)^(2k+m)=(by)^m*(abxy)^2k"

when a、b→1,there is the equation "

(x²+y²-1)^m=y^m"

So we have "|x|^m=((x²+y²-1)/y)^m=1"

Compare plots of "x²+y²-xy=1" and "x²+y²-y=1",we find only when x∈[-1/2,1/2] their difference become obvious,so m→0.

We make m=1，so k→+∞.

By trying,due to the equation's interval,when k=1 we get such a beautiful plot : (x²+y²-1)³=x²y³

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**Sumasoltin****Member**- Registered: 2012-04-28
- Posts: 30

Oh, that's funny ! Without blanks "= (" was transformed into negetive face!

I did use k instead of 5,but a known number looks better.

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**Sumasoltin****Member**- Registered: 2012-04-28
- Posts: 30

(x+13+sqrt((x+4)^2+4)-2sqrt((x+8)^2+4))^2*(12x*y^2+x^3-11y^2+x^2+7x-4)^3+(8y^2+6x^2-16x+4)^3*y^4*(x+13)^2=0 where x=-15 to 5, guess what's in it?

Tip:Thanksgiving Days

*Last edited by Sumasoltin (2012-04-30 01:26:19)*

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,544

Check this out:Some values of k

*Last edited by anonimnystefy (2012-04-30 01:24:49)*

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