Mandelbrot Boundary Area
$begingroup$
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
$endgroup$
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 |
0
active
oldest
votes
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3052209%2fmandelbrot-boundary-area%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3052209%2fmandelbrot-boundary-area%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
$begingroup$
See mathoverflow.net/questions/37229/… and math.stackexchange.com/questions/2594562/…
$endgroup$
– lhf
Dec 25 '18 at 17:20