Connectedness of parts used in the Banach–Tarski paradox
A quote from the Wikipedia article "Axiom of choice":
One example is the Banach–Tarski paradox which says that it is
possible to decompose the 3-dimensional solid unit ball into finitely
many pieces and, using only rotations and translations, reassemble the
pieces into two solid balls each with the same volume as the original.
I know that at least some of the parts (called pieces here) must be non-measurable sets. I wonder if each of them can be chosen to be a path-connected set (otherwise it's really misleading to call them pieces, I think).
measure-theory set-theory axiom-of-choice connectedness paradoxes
asked May 29 '14 at 23:40
Vladimir Reshetnikov
24.2k4119231
add a comment |
A quote from the Wikipedia article "Axiom of choice":
One example is the Banach–Tarski paradox which says that it is
possible to decompose the 3-dimensional solid unit ball into finitely
many pieces and, using only rotations and translations, reassemble the
pieces into two solid balls each with the same volume as the original.
I know that at least some of the parts (called pieces here) must be non-measurable sets. I wonder if each of them can be chosen to be a path-connected set (otherwise it's really misleading to call them pieces, I think).
measure-theory set-theory axiom-of-choice connectedness paradoxes
asked May 29 '14 at 23:40
Vladimir Reshetnikov
24.2k4119231
3
If I recall correctly, one of these pieces is a singleton.
– Asaf Karagila♦
May 29 '14 at 23:43
@AsafKaragila A singleton is a path-connected set, right? Just use a constant function that maps $[0,1]$ to its only point. My question is can we make each part to be path-connected.
– Vladimir Reshetnikov
May 30 '14 at 17:11
1
Yes, I know, which is why I didn't post an answer. I simply don't know the answer. It sounds a bit unlikely, though. In any case, historically Banach-Tarski didn't specify how many pieces; von Neumann claimed he can do it in $9$, and some time later Sierpinski claimed he can do it in $8$; and only some more time later Robinson proved that you can do it in $5$, but you can't do it in $4$. (All this appears in Halbeisen's Combinatorial Set Theory which includes a chapter on the Banach-Tarski paradox.)
– Asaf Karagila♦
May 30 '14 at 17:13
There was an exposition of this result in American Mathematical Monthly. I can't recall the title or year.
– DanielWainfleet
Feb 22 '17 at 4:09
add a comment |
A quote from the Wikipedia article "Axiom of choice":
One example is the Banach–Tarski paradox which says that it is
possible to decompose the 3-dimensional solid unit ball into finitely
many pieces and, using only rotations and translations, reassemble the
pieces into two solid balls each with the same volume as the original.
I know that at least some of the parts (called pieces here) must be non-measurable sets. I wonder if each of them can be chosen to be a path-connected set (otherwise it's really misleading to call them pieces, I think).
measure-theory set-theory axiom-of-choice connectedness paradoxes
asked May 29 '14 at 23:40
Vladimir Reshetnikov
24.2k4119231
A quote from the Wikipedia article "Axiom of choice":
One example is the Banach–Tarski paradox which says that it is
possible to decompose the 3-dimensional solid unit ball into finitely
many pieces and, using only rotations and translations, reassemble the
pieces into two solid balls each with the same volume as the original.
I know that at least some of the parts (called pieces here) must be non-measurable sets. I wonder if each of them can be chosen to be a path-connected set (otherwise it's really misleading to call them pieces, I think).
measure-theory set-theory axiom-of-choice connectedness paradoxes
measure-theory set-theory axiom-of-choice connectedness paradoxes
asked May 29 '14 at 23:40
Vladimir Reshetnikov
24.2k4119231
asked May 29 '14 at 23:40
Vladimir Reshetnikov
24.2k4119231
asked May 29 '14 at 23:40
Vladimir Reshetnikov
24.2k4119231
asked May 29 '14 at 23:40
Vladimir Reshetnikov
24.2k4119231
asked May 29 '14 at 23:40
Vladimir Reshetnikov
24.2k4119231
24.2k4119231
3
If I recall correctly, one of these pieces is a singleton.
– Asaf Karagila♦
May 29 '14 at 23:43
@AsafKaragila A singleton is a path-connected set, right? Just use a constant function that maps $[0,1]$ to its only point. My question is can we make each part to be path-connected.
– Vladimir Reshetnikov
May 30 '14 at 17:11
1
Yes, I know, which is why I didn't post an answer. I simply don't know the answer. It sounds a bit unlikely, though. In any case, historically Banach-Tarski didn't specify how many pieces; von Neumann claimed he can do it in $9$, and some time later Sierpinski claimed he can do it in $8$; and only some more time later Robinson proved that you can do it in $5$, but you can't do it in $4$. (All this appears in Halbeisen's Combinatorial Set Theory which includes a chapter on the Banach-Tarski paradox.)
– Asaf Karagila♦
May 30 '14 at 17:13
There was an exposition of this result in American Mathematical Monthly. I can't recall the title or year.
– DanielWainfleet
Feb 22 '17 at 4:09
add a comment |
3
If I recall correctly, one of these pieces is a singleton.
– Asaf Karagila♦
May 29 '14 at 23:43
@AsafKaragila A singleton is a path-connected set, right? Just use a constant function that maps $[0,1]$ to its only point. My question is can we make each part to be path-connected.
– Vladimir Reshetnikov
May 30 '14 at 17:11
1
Yes, I know, which is why I didn't post an answer. I simply don't know the answer. It sounds a bit unlikely, though. In any case, historically Banach-Tarski didn't specify how many pieces; von Neumann claimed he can do it in $9$, and some time later Sierpinski claimed he can do it in $8$; and only some more time later Robinson proved that you can do it in $5$, but you can't do it in $4$. (All this appears in Halbeisen's Combinatorial Set Theory which includes a chapter on the Banach-Tarski paradox.)
– Asaf Karagila♦
May 30 '14 at 17:13
There was an exposition of this result in American Mathematical Monthly. I can't recall the title or year.
– DanielWainfleet
Feb 22 '17 at 4:09
3
3
If I recall correctly, one of these pieces is a singleton.
– Asaf Karagila♦
May 29 '14 at 23:43
If I recall correctly, one of these pieces is a singleton.
– Asaf Karagila♦
May 29 '14 at 23:43
@AsafKaragila A singleton is a path-connected set, right? Just use a constant function that maps $[0,1]$ to its only point. My question is can we make each part to be path-connected.
– Vladimir Reshetnikov
May 30 '14 at 17:11
@AsafKaragila A singleton is a path-connected set, right? Just use a constant function that maps $[0,1]$ to its only point. My question is can we make each part to be path-connected.
– Vladimir Reshetnikov
May 30 '14 at 17:11
1
1
Yes, I know, which is why I didn't post an answer. I simply don't know the answer. It sounds a bit unlikely, though. In any case, historically Banach-Tarski didn't specify how many pieces; von Neumann claimed he can do it in $9$, and some time later Sierpinski claimed he can do it in $8$; and only some more time later Robinson proved that you can do it in $5$, but you can't do it in $4$. (All this appears in Halbeisen's Combinatorial Set Theory which includes a chapter on the Banach-Tarski paradox.)
– Asaf Karagila♦
May 30 '14 at 17:13
Yes, I know, which is why I didn't post an answer. I simply don't know the answer. It sounds a bit unlikely, though. In any case, historically Banach-Tarski didn't specify how many pieces; von Neumann claimed he can do it in $9$, and some time later Sierpinski claimed he can do it in $8$; and only some more time later Robinson proved that you can do it in $5$, but you can't do it in $4$. (All this appears in Halbeisen's Combinatorial Set Theory which includes a chapter on the Banach-Tarski paradox.)
– Asaf Karagila♦
May 30 '14 at 17:13
There was an exposition of this result in American Mathematical Monthly. I can't recall the title or year.
– DanielWainfleet
Feb 22 '17 at 4:09
There was an exposition of this result in American Mathematical Monthly. I can't recall the title or year.
– DanielWainfleet
Feb 22 '17 at 4:09
add a comment |
1 Answer
1
active
oldest
votes
Dekker and de Groot proved (Decompositions of a sphere, Fund. Math. 43 (1956), 185–194) that the pieces can be made to be totally imperfect, and hence connected and locally connected. I don't know about path connected, though.
Although you didn't ask about this, one can also argue that "decompose and reassemble" implicitly suggests that the rearrangement can take place using continuous motions during which the pieces never intersect. That this is possible was proved by Trevor Wilson
(A continuous movement version of the Banach–Tarski paradox: A solution to de Groot's problem, J. Symb. Logic 70 (2005), 946–952). But I don't know if there is a common generalization of the Dekker–de Groot result above and Wilson's result.
answered Dec 9 at 0:53
Timothy Chow
17613
add a comment |
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%2f814355%2fconnectedness-of-parts-used-in-the-banach-tarski-paradox%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
Dekker and de Groot proved (Decompositions of a sphere, Fund. Math. 43 (1956), 185–194) that the pieces can be made to be totally imperfect, and hence connected and locally connected. I don't know about path connected, though.
Although you didn't ask about this, one can also argue that "decompose and reassemble" implicitly suggests that the rearrangement can take place using continuous motions during which the pieces never intersect. That this is possible was proved by Trevor Wilson
(A continuous movement version of the Banach–Tarski paradox: A solution to de Groot's problem, J. Symb. Logic 70 (2005), 946–952). But I don't know if there is a common generalization of the Dekker–de Groot result above and Wilson's result.
answered Dec 9 at 0:53
Timothy Chow
17613
add a comment |
Dekker and de Groot proved (Decompositions of a sphere, Fund. Math. 43 (1956), 185–194) that the pieces can be made to be totally imperfect, and hence connected and locally connected. I don't know about path connected, though.
Although you didn't ask about this, one can also argue that "decompose and reassemble" implicitly suggests that the rearrangement can take place using continuous motions during which the pieces never intersect. That this is possible was proved by Trevor Wilson
(A continuous movement version of the Banach–Tarski paradox: A solution to de Groot's problem, J. Symb. Logic 70 (2005), 946–952). But I don't know if there is a common generalization of the Dekker–de Groot result above and Wilson's result.
answered Dec 9 at 0:53
Timothy Chow
17613
add a comment |
Dekker and de Groot proved (Decompositions of a sphere, Fund. Math. 43 (1956), 185–194) that the pieces can be made to be totally imperfect, and hence connected and locally connected. I don't know about path connected, though.
Although you didn't ask about this, one can also argue that "decompose and reassemble" implicitly suggests that the rearrangement can take place using continuous motions during which the pieces never intersect. That this is possible was proved by Trevor Wilson
(A continuous movement version of the Banach–Tarski paradox: A solution to de Groot's problem, J. Symb. Logic 70 (2005), 946–952). But I don't know if there is a common generalization of the Dekker–de Groot result above and Wilson's result.
answered Dec 9 at 0:53
Timothy Chow
17613
Dekker and de Groot proved (Decompositions of a sphere, Fund. Math. 43 (1956), 185–194) that the pieces can be made to be totally imperfect, and hence connected and locally connected. I don't know about path connected, though.
Although you didn't ask about this, one can also argue that "decompose and reassemble" implicitly suggests that the rearrangement can take place using continuous motions during which the pieces never intersect. That this is possible was proved by Trevor Wilson
(A continuous movement version of the Banach–Tarski paradox: A solution to de Groot's problem, J. Symb. Logic 70 (2005), 946–952). But I don't know if there is a common generalization of the Dekker–de Groot result above and Wilson's result.
answered Dec 9 at 0:53
Timothy Chow
17613
answered Dec 9 at 0:53
Timothy Chow
17613
answered Dec 9 at 0:53
Timothy Chow
17613
answered Dec 9 at 0:53
Timothy Chow
17613
17613
add a comment |
add a comment |
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.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- 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.
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%2f814355%2fconnectedness-of-parts-used-in-the-banach-tarski-paradox%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
3
If I recall correctly, one of these pieces is a singleton.
– Asaf Karagila♦
May 29 '14 at 23:43
@AsafKaragila A singleton is a path-connected set, right? Just use a constant function that maps $[0,1]$ to its only point. My question is can we make each part to be path-connected.
– Vladimir Reshetnikov
May 30 '14 at 17:11
1
Yes, I know, which is why I didn't post an answer. I simply don't know the answer. It sounds a bit unlikely, though. In any case, historically Banach-Tarski didn't specify how many pieces; von Neumann claimed he can do it in $9$, and some time later Sierpinski claimed he can do it in $8$; and only some more time later Robinson proved that you can do it in $5$, but you can't do it in $4$. (All this appears in Halbeisen's Combinatorial Set Theory which includes a chapter on the Banach-Tarski paradox.)
– Asaf Karagila♦
May 30 '14 at 17:13
There was an exposition of this result in American Mathematical Monthly. I can't recall the title or year.
– DanielWainfleet
Feb 22 '17 at 4:09