Triangles with Congruent Corresponding Sides that Cannot fold into a Tetrahedron - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T19:05:24Z http://mathoverflow.net/feeds/question/89802 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/89802/triangles-with-congruent-corresponding-sides-that-cannot-fold-into-a-tetrahedron Triangles with Congruent Corresponding Sides that Cannot fold into a Tetrahedron drum 2012-02-28T22:22:40Z 2012-05-28T06:33:48Z <p>I've been trying to find, without much success, 4 triangles whose corresponding sides are congruent that cannot be folded into a tetrahedron.</p> <p>Anyone has any clue how to approach this problem?</p> http://mathoverflow.net/questions/89802/triangles-with-congruent-corresponding-sides-that-cannot-fold-into-a-tetrahedron/89803#89803 Answer by Igor Rivin for Triangles with Congruent Corresponding Sides that Cannot fold into a Tetrahedron Igor Rivin 2012-02-28T22:34:28Z 2012-02-28T22:34:28Z <p>What do you mean by "corresponding sides"? If what you mean that you have a gluing diagram which is consistent, just take your triangles $ABC, ABD, ACD, BCD$ in such a way that the angles at $A$ in all three triangles sharing that vertex is $5\pi/6,$ and otherwise the three triangles with vertex at $A$ are isosceles (so the other two angles are $\pi/10$) the triangle $BCD$ is equilateral. Notice that these triangles do not glue into a tetrahedron, since the total angle at $A$ is greater than $2\pi$ (since $15/6 > 2$).</p> http://mathoverflow.net/questions/89802/triangles-with-congruent-corresponding-sides-that-cannot-fold-into-a-tetrahedron/98173#98173 Answer by Carl Sebeny for Triangles with Congruent Corresponding Sides that Cannot fold into a Tetrahedron Carl Sebeny 2012-05-28T06:33:48Z 2012-05-28T06:33:48Z <p>A tetrahedron with congruent faces will have all acute face angles. No obtuse or right angles. </p>