Every Strictly-Convex Deltahedron

The video begins with an introduction to the diversity of shapes, specifically focusing on the icosahedron and its 20 equilateral triangular faces. It then compares this shape to other platonic solids made of equilateral triangles, such as the tetrahedron with its four faces. The process of creating a tetrahedron from a net of triangles is demonstrated, humorously referred to as 'The Tetrahedral Hug.' The video also introduces the concept of deltahedra, solids composed entirely of equilateral triangles, and mentions that there are eight strictly convex deltahedra. Pyramids and prisms are explained, with examples of square and pentagonal pyramids, and the special case of a cube as a square prism. The video concludes this section by discussing the connection of two parallel polygons using equilateral triangles to form a square antiprism.

The video moves on to discuss the processes of elongation and gyroelongation, which are analogous to constructing prisms and antiprisms but involve connecting solids instead of polygons. An octahedron is used as an example to demonstrate how to create an elongated square bipyramid by separating two square pyramids and inserting squares in between. The concept of gyroelongation is introduced by replacing squares with equilateral triangles to form a gyroelongated square bipyramid, which is identified as a strictly convex deltahedron. The video also attempts to gyroelongate a triangular bipyramid but finds that the result is not strictly convex, leading to the creation of a rhombohedron. The process is further illustrated by gyroelongating a pentagonal pyramid to form an icosahedron, which is also a strictly convex deltahedron. The video concludes with the construction of a pentagonal bipyramid, which is another example of a strictly convex deltahedron.

The video continues with the geometric concept of augmentation, where a shape's face is replaced with a solid. Using a triangular prism as an example, the video shows how to replace square faces with equilateral triangles to create a triaugmented triangular prism, which is recognized as the seventh strictly convex deltahedron. The final shape to be constructed is the snub disphenoid, which has a complex name explained in the video. It is created by starting with a square antiprism and replacing the top and base squares with equilateral triangles, resulting in a shape with two mirror-image versions. The video explains that the snub disphenoid is the eighth and final strictly convex deltahedron that can exist.

The term 'disphenoid,' meaning 'wedge shape,' is explored, noting that a regular tetrahedron is a type of disphenoid. The video demonstrates how to construct a tetrahedron using isosceles triangles and how the term 'disphenoid' becomes more visually apparent with non-equilateral triangles. The concept of expansion in geometry is introduced using a cube, where each face is moved away from the center to form an expanded cube. The gaps created are filled with squares and equilateral triangles to form a new shape. The video then transitions to the concept of snubification, which involves expanding the faces of a shape and connecting them with equilateral triangles. This process, when applied to a tetrahedron (a disphenoid), results in a snub disphenoid, thus explaining the origin of its name.

In the final paragraph, the video humorously addresses the abundance of squares used throughout the construction of various shapes, referring to them as 'Whole Bunch O' Squares™.' The squares are playfully indicated to be kept on a 'Trophy Shelf™,' emphasizing the value placed on being 'square' or a nerd in the context of the video's theme. The host pokes fun at the stereotype of a nerd being obsessed with shapes and ends the video with a light-hearted acknowledgment of their own enthusiasm for the subject.