Mandelbrot Boundary Area
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I might be entirely off an my assumptions, but the following has led me to a question.
The Mandelbrot set is contained by an border of infinite length.
Said border is 2-dimensional.
The Hilbert space-filling curve also is 2-dimensional and infinite in length.
The Hilbert curve covers an area of 1 square unit.
Can the Mandelbrot border be assigned an area? Am I justified in saying that the Hilbert curve is has area 1? If it is two dimensional, that seems reasonable. What terminology is used for non-integer dimensions? Also, for integer dimensions. What do you call the measure of the 4D interior of a tesseract? 5D?
I'm in 10th grade, but very good at math (taking Calc BC), so as long as answers don't go to deep into metric spaces, measure theory, topology, et cetera, I should be able to follow them.
Thanks in advance!
measure-theory fractals dimension-theory
asked Dec 25 '18 at 15:40
William GrannisWilliam Grannis
1,030521
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add a comment |
$begingroup$
I might be entirely off an my assumptions, but the following has led me to a question.
The Mandelbrot set is contained by an border of infinite length.
Said border is 2-dimensional.
The Hilbert space-filling curve also is 2-dimensional and infinite in length.
The Hilbert curve covers an area of 1 square unit.
Can the Mandelbrot border be assigned an area? Am I justified in saying that the Hilbert curve is has area 1? If it is two dimensional, that seems reasonable. What terminology is used for non-integer dimensions? Also, for integer dimensions. What do you call the measure of the 4D interior of a tesseract? 5D?
I'm in 10th grade, but very good at math (taking Calc BC), so as long as answers don't go to deep into metric spaces, measure theory, topology, et cetera, I should be able to follow them.
Thanks in advance!
measure-theory fractals dimension-theory
asked Dec 25 '18 at 15:40
William GrannisWilliam Grannis
1,030521
$endgroup$
$begingroup$
See mathoverflow.net/questions/37229/… and math.stackexchange.com/questions/2594562/…
$endgroup$
– lhf
Dec 25 '18 at 17:20
add a comment |
$begingroup$
I might be entirely off an my assumptions, but the following has led me to a question.
The Mandelbrot set is contained by an border of infinite length.
Said border is 2-dimensional.
The Hilbert space-filling curve also is 2-dimensional and infinite in length.
The Hilbert curve covers an area of 1 square unit.
Can the Mandelbrot border be assigned an area? Am I justified in saying that the Hilbert curve is has area 1? If it is two dimensional, that seems reasonable. What terminology is used for non-integer dimensions? Also, for integer dimensions. What do you call the measure of the 4D interior of a tesseract? 5D?
I'm in 10th grade, but very good at math (taking Calc BC), so as long as answers don't go to deep into metric spaces, measure theory, topology, et cetera, I should be able to follow them.
Thanks in advance!
measure-theory fractals dimension-theory
asked Dec 25 '18 at 15:40
William GrannisWilliam Grannis
1,030521
$endgroup$
I might be entirely off an my assumptions, but the following has led me to a question.
The Mandelbrot set is contained by an border of infinite length.
Said border is 2-dimensional.
The Hilbert space-filling curve also is 2-dimensional and infinite in length.
The Hilbert curve covers an area of 1 square unit.
Can the Mandelbrot border be assigned an area? Am I justified in saying that the Hilbert curve is has area 1? If it is two dimensional, that seems reasonable. What terminology is used for non-integer dimensions? Also, for integer dimensions. What do you call the measure of the 4D interior of a tesseract? 5D?
I'm in 10th grade, but very good at math (taking Calc BC), so as long as answers don't go to deep into metric spaces, measure theory, topology, et cetera, I should be able to follow them.
Thanks in advance!
measure-theory fractals dimension-theory
measure-theory fractals dimension-theory
asked Dec 25 '18 at 15:40
William GrannisWilliam Grannis
1,030521
asked Dec 25 '18 at 15:40
William GrannisWilliam Grannis
1,030521
asked Dec 25 '18 at 15:40
William GrannisWilliam Grannis
1,030521
asked Dec 25 '18 at 15:40
William GrannisWilliam Grannis
1,030521
asked Dec 25 '18 at 15:40
William GrannisWilliam Grannis
1,030521
1,030521
$begingroup$
See mathoverflow.net/questions/37229/… and math.stackexchange.com/questions/2594562/…
$endgroup$
– lhf
Dec 25 '18 at 17:20
add a comment |
$begingroup$
See mathoverflow.net/questions/37229/… and math.stackexchange.com/questions/2594562/…
$endgroup$
– lhf
Dec 25 '18 at 17:20
$begingroup$
See mathoverflow.net/questions/37229/… and math.stackexchange.com/questions/2594562/…
$endgroup$
– lhf
Dec 25 '18 at 17:20
$begingroup$
See mathoverflow.net/questions/37229/… and math.stackexchange.com/questions/2594562/…
$endgroup$
– lhf
Dec 25 '18 at 17:20
add a comment |
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$begingroup$
See mathoverflow.net/questions/37229/… and math.stackexchange.com/questions/2594562/…
$endgroup$
– lhf
Dec 25 '18 at 17:20