Octahedra with four equilateral faces












1












$begingroup$


Let $A_1, A_2, A_3, A'_1, A'_2, A'_3$ be the vertices of a (not-necessarily convex) octahedron; here $X'$ is the vertex not on an edge with $X$. Suppose that the four non-adjacent triangular faces $A_1 A_2 A_3$, $A_1 A'_2 A'_3$,$A'_1 A_2 A'_3$,$A'_1 A'_2 A_3$ are equilateral with sides of lengths $x_0, x_1, x_2, x_3$.



I am interested in knowing the conditions on $x_0,x_1,x_2,x_3$ that must be satisfied for such a configuration to exist; and how unique the configuration is. This is a preliminary step to understanding this Question on Octahedra
and the answer by @Blue.



Clearly it is necessary that $x_i>0$ for each $i$; and by using the triangle inequality on the other four faces, that $x_i+x_j>x_k$ for every 3-set ${i,j,k}subset {0,1,2,3}$.



Question 1: are these conditions sufficient for existence? I think so, but have only a hand-waving argument and would like something better.



It seems to me that if such a configuration exists for some $x_0,x_1,x_2,x_3$ then in general if we fix $A_2, A_3, A'_1, A'_2, A'_3$ there are two points $A_1$ which satisfy the right conditions, giving a convex octahedron and a non-convex one. Again I can only wave my hands, and my intuition may be wrong. So,



Question 2: what can be proved about uniqueness?



Lastly, to get some feeling for which $(x_0,x_1, x_2,x_3)$ yield an octahedron, I would like to understand their geometry.



Question 3: Describe geometrically the (subset of) the solid bounded by



begin{align*}
x_0+x_1+x_2+x_3&=1;&\
x_i&=0, &i=0,dots,3; \
x_i+x_j&=x_k &textrm{ for every 3-set } {i,j,k}subset {0,1,2,3}
end{align*}



which yield solutions.










share|cite|improve this question









$endgroup$

















    1












    $begingroup$


    Let $A_1, A_2, A_3, A'_1, A'_2, A'_3$ be the vertices of a (not-necessarily convex) octahedron; here $X'$ is the vertex not on an edge with $X$. Suppose that the four non-adjacent triangular faces $A_1 A_2 A_3$, $A_1 A'_2 A'_3$,$A'_1 A_2 A'_3$,$A'_1 A'_2 A_3$ are equilateral with sides of lengths $x_0, x_1, x_2, x_3$.



    I am interested in knowing the conditions on $x_0,x_1,x_2,x_3$ that must be satisfied for such a configuration to exist; and how unique the configuration is. This is a preliminary step to understanding this Question on Octahedra
    and the answer by @Blue.



    Clearly it is necessary that $x_i>0$ for each $i$; and by using the triangle inequality on the other four faces, that $x_i+x_j>x_k$ for every 3-set ${i,j,k}subset {0,1,2,3}$.



    Question 1: are these conditions sufficient for existence? I think so, but have only a hand-waving argument and would like something better.



    It seems to me that if such a configuration exists for some $x_0,x_1,x_2,x_3$ then in general if we fix $A_2, A_3, A'_1, A'_2, A'_3$ there are two points $A_1$ which satisfy the right conditions, giving a convex octahedron and a non-convex one. Again I can only wave my hands, and my intuition may be wrong. So,



    Question 2: what can be proved about uniqueness?



    Lastly, to get some feeling for which $(x_0,x_1, x_2,x_3)$ yield an octahedron, I would like to understand their geometry.



    Question 3: Describe geometrically the (subset of) the solid bounded by



    begin{align*}
    x_0+x_1+x_2+x_3&=1;&\
    x_i&=0, &i=0,dots,3; \
    x_i+x_j&=x_k &textrm{ for every 3-set } {i,j,k}subset {0,1,2,3}
    end{align*}



    which yield solutions.










    share|cite|improve this question









    $endgroup$















      1












      1








      1





      $begingroup$


      Let $A_1, A_2, A_3, A'_1, A'_2, A'_3$ be the vertices of a (not-necessarily convex) octahedron; here $X'$ is the vertex not on an edge with $X$. Suppose that the four non-adjacent triangular faces $A_1 A_2 A_3$, $A_1 A'_2 A'_3$,$A'_1 A_2 A'_3$,$A'_1 A'_2 A_3$ are equilateral with sides of lengths $x_0, x_1, x_2, x_3$.



      I am interested in knowing the conditions on $x_0,x_1,x_2,x_3$ that must be satisfied for such a configuration to exist; and how unique the configuration is. This is a preliminary step to understanding this Question on Octahedra
      and the answer by @Blue.



      Clearly it is necessary that $x_i>0$ for each $i$; and by using the triangle inequality on the other four faces, that $x_i+x_j>x_k$ for every 3-set ${i,j,k}subset {0,1,2,3}$.



      Question 1: are these conditions sufficient for existence? I think so, but have only a hand-waving argument and would like something better.



      It seems to me that if such a configuration exists for some $x_0,x_1,x_2,x_3$ then in general if we fix $A_2, A_3, A'_1, A'_2, A'_3$ there are two points $A_1$ which satisfy the right conditions, giving a convex octahedron and a non-convex one. Again I can only wave my hands, and my intuition may be wrong. So,



      Question 2: what can be proved about uniqueness?



      Lastly, to get some feeling for which $(x_0,x_1, x_2,x_3)$ yield an octahedron, I would like to understand their geometry.



      Question 3: Describe geometrically the (subset of) the solid bounded by



      begin{align*}
      x_0+x_1+x_2+x_3&=1;&\
      x_i&=0, &i=0,dots,3; \
      x_i+x_j&=x_k &textrm{ for every 3-set } {i,j,k}subset {0,1,2,3}
      end{align*}



      which yield solutions.










      share|cite|improve this question









      $endgroup$




      Let $A_1, A_2, A_3, A'_1, A'_2, A'_3$ be the vertices of a (not-necessarily convex) octahedron; here $X'$ is the vertex not on an edge with $X$. Suppose that the four non-adjacent triangular faces $A_1 A_2 A_3$, $A_1 A'_2 A'_3$,$A'_1 A_2 A'_3$,$A'_1 A'_2 A_3$ are equilateral with sides of lengths $x_0, x_1, x_2, x_3$.



      I am interested in knowing the conditions on $x_0,x_1,x_2,x_3$ that must be satisfied for such a configuration to exist; and how unique the configuration is. This is a preliminary step to understanding this Question on Octahedra
      and the answer by @Blue.



      Clearly it is necessary that $x_i>0$ for each $i$; and by using the triangle inequality on the other four faces, that $x_i+x_j>x_k$ for every 3-set ${i,j,k}subset {0,1,2,3}$.



      Question 1: are these conditions sufficient for existence? I think so, but have only a hand-waving argument and would like something better.



      It seems to me that if such a configuration exists for some $x_0,x_1,x_2,x_3$ then in general if we fix $A_2, A_3, A'_1, A'_2, A'_3$ there are two points $A_1$ which satisfy the right conditions, giving a convex octahedron and a non-convex one. Again I can only wave my hands, and my intuition may be wrong. So,



      Question 2: what can be proved about uniqueness?



      Lastly, to get some feeling for which $(x_0,x_1, x_2,x_3)$ yield an octahedron, I would like to understand their geometry.



      Question 3: Describe geometrically the (subset of) the solid bounded by



      begin{align*}
      x_0+x_1+x_2+x_3&=1;&\
      x_i&=0, &i=0,dots,3; \
      x_i+x_j&=x_k &textrm{ for every 3-set } {i,j,k}subset {0,1,2,3}
      end{align*}



      which yield solutions.







      geometry polyhedra






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Dec 26 '18 at 13:16









      ancientmathematicianancientmathematician

      4,4481513




      4,4481513






















          1 Answer
          1






          active

          oldest

          votes


















          1












          $begingroup$

          This is not an answer, but just a conjecture about Question 1.



          Consider your octahedron "unfolded" on a plane, and in particular its equilateral face $ABC$ of side $x_0$ and its three neighbouring (not equilateral) faces $ABE$, $BCD$, $ACF$ (see diagrams below). We can imagine to "fold" these neighbouring faces (all on the same side of the equilateral face) until $DE=DB$, $DF=CF$ and $FE=AE$. We have then three equations for three unknown parameters (the positions of points $D$, $E$, $F$ along the three circles where they are constrained to lie).



          But if you try that with GeoGebra, it turns out that the construction succeeds only if, in the unfolded planar graph, $DEge DB$, $DFge CF$ and $FEge AE$. Thus the first diagram shows a constructible octahedron, while the second one is not constructible, because $DE<DB$. I haven't a proof for that (yet) and would like to know other opinions on this conjecture.



          enter image description hereenter image description here



          A short comment on Question 3: I think the first condition can always be met by a suitable rescaling, the second one gives a single point and the third one (as written) implies the second one. But maybe I just don't understand what you are asking.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            Yes, I think I was mentally folding something that wouldn't join up, I must look again. As to Qn 3: sure, the interesting set is really projective, and eqn 1 is just a way of getting into $mathbb{R}^3$: then we have $4+12$ bounding plane "faces". Plus whatever other restrictions there are.
            $endgroup$
            – ancientmathematician
            Jan 2 at 7:53













          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
          });


          }
          });














          draft saved

          draft discarded


















          StackExchange.ready(
          function () {
          StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3052935%2foctahedra-with-four-equilateral-faces%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









          1












          $begingroup$

          This is not an answer, but just a conjecture about Question 1.



          Consider your octahedron "unfolded" on a plane, and in particular its equilateral face $ABC$ of side $x_0$ and its three neighbouring (not equilateral) faces $ABE$, $BCD$, $ACF$ (see diagrams below). We can imagine to "fold" these neighbouring faces (all on the same side of the equilateral face) until $DE=DB$, $DF=CF$ and $FE=AE$. We have then three equations for three unknown parameters (the positions of points $D$, $E$, $F$ along the three circles where they are constrained to lie).



          But if you try that with GeoGebra, it turns out that the construction succeeds only if, in the unfolded planar graph, $DEge DB$, $DFge CF$ and $FEge AE$. Thus the first diagram shows a constructible octahedron, while the second one is not constructible, because $DE<DB$. I haven't a proof for that (yet) and would like to know other opinions on this conjecture.



          enter image description hereenter image description here



          A short comment on Question 3: I think the first condition can always be met by a suitable rescaling, the second one gives a single point and the third one (as written) implies the second one. But maybe I just don't understand what you are asking.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            Yes, I think I was mentally folding something that wouldn't join up, I must look again. As to Qn 3: sure, the interesting set is really projective, and eqn 1 is just a way of getting into $mathbb{R}^3$: then we have $4+12$ bounding plane "faces". Plus whatever other restrictions there are.
            $endgroup$
            – ancientmathematician
            Jan 2 at 7:53


















          1












          $begingroup$

          This is not an answer, but just a conjecture about Question 1.



          Consider your octahedron "unfolded" on a plane, and in particular its equilateral face $ABC$ of side $x_0$ and its three neighbouring (not equilateral) faces $ABE$, $BCD$, $ACF$ (see diagrams below). We can imagine to "fold" these neighbouring faces (all on the same side of the equilateral face) until $DE=DB$, $DF=CF$ and $FE=AE$. We have then three equations for three unknown parameters (the positions of points $D$, $E$, $F$ along the three circles where they are constrained to lie).



          But if you try that with GeoGebra, it turns out that the construction succeeds only if, in the unfolded planar graph, $DEge DB$, $DFge CF$ and $FEge AE$. Thus the first diagram shows a constructible octahedron, while the second one is not constructible, because $DE<DB$. I haven't a proof for that (yet) and would like to know other opinions on this conjecture.



          enter image description hereenter image description here



          A short comment on Question 3: I think the first condition can always be met by a suitable rescaling, the second one gives a single point and the third one (as written) implies the second one. But maybe I just don't understand what you are asking.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            Yes, I think I was mentally folding something that wouldn't join up, I must look again. As to Qn 3: sure, the interesting set is really projective, and eqn 1 is just a way of getting into $mathbb{R}^3$: then we have $4+12$ bounding plane "faces". Plus whatever other restrictions there are.
            $endgroup$
            – ancientmathematician
            Jan 2 at 7:53
















          1












          1








          1





          $begingroup$

          This is not an answer, but just a conjecture about Question 1.



          Consider your octahedron "unfolded" on a plane, and in particular its equilateral face $ABC$ of side $x_0$ and its three neighbouring (not equilateral) faces $ABE$, $BCD$, $ACF$ (see diagrams below). We can imagine to "fold" these neighbouring faces (all on the same side of the equilateral face) until $DE=DB$, $DF=CF$ and $FE=AE$. We have then three equations for three unknown parameters (the positions of points $D$, $E$, $F$ along the three circles where they are constrained to lie).



          But if you try that with GeoGebra, it turns out that the construction succeeds only if, in the unfolded planar graph, $DEge DB$, $DFge CF$ and $FEge AE$. Thus the first diagram shows a constructible octahedron, while the second one is not constructible, because $DE<DB$. I haven't a proof for that (yet) and would like to know other opinions on this conjecture.



          enter image description hereenter image description here



          A short comment on Question 3: I think the first condition can always be met by a suitable rescaling, the second one gives a single point and the third one (as written) implies the second one. But maybe I just don't understand what you are asking.






          share|cite|improve this answer









          $endgroup$



          This is not an answer, but just a conjecture about Question 1.



          Consider your octahedron "unfolded" on a plane, and in particular its equilateral face $ABC$ of side $x_0$ and its three neighbouring (not equilateral) faces $ABE$, $BCD$, $ACF$ (see diagrams below). We can imagine to "fold" these neighbouring faces (all on the same side of the equilateral face) until $DE=DB$, $DF=CF$ and $FE=AE$. We have then three equations for three unknown parameters (the positions of points $D$, $E$, $F$ along the three circles where they are constrained to lie).



          But if you try that with GeoGebra, it turns out that the construction succeeds only if, in the unfolded planar graph, $DEge DB$, $DFge CF$ and $FEge AE$. Thus the first diagram shows a constructible octahedron, while the second one is not constructible, because $DE<DB$. I haven't a proof for that (yet) and would like to know other opinions on this conjecture.



          enter image description hereenter image description here



          A short comment on Question 3: I think the first condition can always be met by a suitable rescaling, the second one gives a single point and the third one (as written) implies the second one. But maybe I just don't understand what you are asking.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Jan 1 at 18:08









          AretinoAretino

          24k21443




          24k21443












          • $begingroup$
            Yes, I think I was mentally folding something that wouldn't join up, I must look again. As to Qn 3: sure, the interesting set is really projective, and eqn 1 is just a way of getting into $mathbb{R}^3$: then we have $4+12$ bounding plane "faces". Plus whatever other restrictions there are.
            $endgroup$
            – ancientmathematician
            Jan 2 at 7:53




















          • $begingroup$
            Yes, I think I was mentally folding something that wouldn't join up, I must look again. As to Qn 3: sure, the interesting set is really projective, and eqn 1 is just a way of getting into $mathbb{R}^3$: then we have $4+12$ bounding plane "faces". Plus whatever other restrictions there are.
            $endgroup$
            – ancientmathematician
            Jan 2 at 7:53


















          $begingroup$
          Yes, I think I was mentally folding something that wouldn't join up, I must look again. As to Qn 3: sure, the interesting set is really projective, and eqn 1 is just a way of getting into $mathbb{R}^3$: then we have $4+12$ bounding plane "faces". Plus whatever other restrictions there are.
          $endgroup$
          – ancientmathematician
          Jan 2 at 7:53






          $begingroup$
          Yes, I think I was mentally folding something that wouldn't join up, I must look again. As to Qn 3: sure, the interesting set is really projective, and eqn 1 is just a way of getting into $mathbb{R}^3$: then we have $4+12$ bounding plane "faces". Plus whatever other restrictions there are.
          $endgroup$
          – ancientmathematician
          Jan 2 at 7:53




















          draft saved

          draft discarded




















































          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.




          draft saved


          draft discarded














          StackExchange.ready(
          function () {
          StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3052935%2foctahedra-with-four-equilateral-faces%23new-answer', 'question_page');
          }
          );

          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







          Popular posts from this blog

          Ellipse (mathématiques)

          Quarter-circle Tiles

          Mont Emei