Quaternion

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Quaternion Julia Fractals
compiled by Michael Bailey
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Complex Numbers
ah bi where a,b are in
R lta,bgt h1, iv-1
Certain special properties exist for complex
numbers that make them extremely useful. Let
X,Y,Z be in C. Also, define 0 as lt0,0gt.
Multiplication
Addition
Commutativity Associativity Distributivity Iden
tity Inverse
XY YX X(YZ) (XY)Z X(YZ)
XYXZ X0X0X X(-X)0(-X)X
XY YX X(YZ) (XY)Z (XY)Z XZYZ 1X X
X1 XX 1 XX
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Complex Number Uses
The properties on the previous page are called
the field axioms, and sets that follow them are
called fields. Because the set of complex
numbers form a field over addition and
multiplication, there are A LOT of practical uses
for them. Some areas that use them
are Quantum Mechanics Relativity Fluid
Dynamics ...
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...and of course fractals!
Julia Set f(Z) Z² A where Z,A are in C
As we learned in PHY307, Iterating the complex
function f, and plotting on complex plane we can
generate complex julia fractals.
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Fractal Dimension
Most of the fractals we studied were between
dimensions of 1 and 2, exclusive.
Julia Set
Sierpinski triangle
Dimension d1.585
Dimension d in (1,2)
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Higher Dimensional Fractals
Sierpinski Pyramid
Sierpinski Pyramid
?
Dimension d2.0
Dimension d in (2,3)
Is there a 3D version of the Julia Set?
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No 3D Julia Equivalent
No. The reason why involves a deep understanding
of group theory which is beyond the scope of this
presentation. However ... ALL IS NOT
LOST! 4D Julia fractals do exist which
utilize quaternions An SAT/GRE style
analogy 2D Julia fractals Complex set
4D Julia fractals Quaternion set
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Quaternion Numbers
ah bi cj dk where a,b,c,d are in
R lta,b,c,dgt h1, i²j²k²ijk-1
Certain special properties exist for quaternions
numbers too. Let X,Y,Z be in H. Also, define 0 as
lt0,0,0,0gt.
However, multiplicative commutativity is not one
of them! Therefore, they're not a field! (They're
a division algebra)
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Quaternions Uses
..Therefore, their applicability is vastly
limited. However, there are some areas that make
heavy use of quaternions. Great for 3D
rotations ... used heavily in 3D
graphics Maxwell's Equations ... originally
anyway!
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Visualizing Quaternion Julia Fractals
Quaternion Julia Fractals use the same type of
generating function, f(Z) Z² A. Except that
Z,A are in H instead of C. So, in order to
visualize these beasts, we need 4 dimensions.
Originally I planned on rendering 3D timeslices
with OpenGL and gradually evolve the time axis
however, with my limited computational power this
is infeasible. So instead I render 2D slices of
3D timeslices, rotating the 3D object in 3
spatial dimension, while evolving the time axis
to create an animation.