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...

The best-known developments of a regular tetrahedron are an equilateral triangle and a parallelogram. Are there any other convex developments of a regular tetrahedron? In this paper we will show that there are convex developments of a regular tetrahedron having the following shapes: an equilateral triangle, an isosceles triangle, a right-angled triangle, scalene triangles, rectangles, parallelo...

The topotactic connection style of oxygen octahedron/tetrahedron in transition metal oxides (TMOs) is an important feature that modulates their corresponding physical properties. Using a simple chemical doping technique, we obtained Sr1−xNdxCoO3-δ with crystal structure from face-shared octahedron to corner-shared octahedron/tetrahedron. Rietveld analyses the X-ray diffraction (XRD) patterns sh...