A Regular Tetrahedron Passes through a Hole Smaller than Its Face
نویسندگان
چکیده
We prove that no triangular frame can hold a convex body, and a convex body can pass through a triangular hole ∆ if and only if the convex body can be congruently embedded in a right triangular prism with base ∆. Applying these result, we prove that a regular tetrahedron of unit edge can pass through an equilateral triangular hole if and only if the edge length of the hole is at least (1 + √ 2)/ √ 6 ≈ 0.9856.
منابع مشابه
Tetrahedra Passing through a Triangular Hole
We prove an embedding theorem that says a convex body can pass through a triangular hole∆ if and only if the convex body can be congruently embedded in a right triangular prism with base ∆. Combining this with a known result on congruent embeddings of a regular tetrahedron in a triangular prism, we show that a regular tetrahedron with unit edge can pass through an equilateral triangular hole in...
متن کاملTetrahedra passing through a triangular hole, and tetrahedra fixed by a planar frame
We show that a convex body can pass through a triangular hole iff it can do so by a translation along a line perpendicular to the hole. As an application, we determine the minimum size of an equilateral triangular hole through which a regular tetrahedron with unit edge can pass. The minimum edge length of the hole is (1+ √ 2)/ √ 6 ≈ 0.9856. One of the key facts for the proof is that no triangul...
متن کاملRegular Simplices Passing through Holes
What is the smallest circular or square wall hole that a regular tetrahedron can pass? This problem was solved by Itoh–Tanoue– Zamfirescu [8]. Then, we settled the case of equilateral triangular hole in [1]. Motivated by these results, we consider the corresponding problems in higher dimensions. Among other results, we determine the minimum (n−1)-dimensional ball hole that a unit regular n-simp...
متن کاملA Characterization by Optimization of the Monge Point of a Tetrahedron
“... nihil omnino in mundo contingint, in quo non maximi minimive ratio quapiam eluceat”, translated into “... nothing in all the world will occur in which no maximum or minimum rule is somehow shining forth”, used to say L.Euler in 1744. This is confirmed by numerous applications of mathematics in physics, mechanics, economy, etc. In this note, we show that it is also the case for the classica...
متن کاملConvex bodies passing through holes
For a given convex body, find a “small” wall hole through which the convex body can pass. This type of problems goes back to Zindler [14] in 1920, who considered a convex polytope which can pass through a fairly small circular holes. A related topic known as Prince Rupert’s problem can be found in [2]. Here we concentrate on the case when the convex body is a regular tetrahedron or a regular n-...
متن کامل