The first chapter, Perpendiculars and Parallels (II), of the ninth unit in this SMSG series includes a discussion of the properties of triangles, circles and perpendiculars, parallels in space, perpendicular lines and planes, and parallel planes. The next chapter, on coordinate geometry, covers distance; midpoints; algebraic descriptions of…

A mathematical derivation is given, developing the cardioid as an epicycloid locus. Curve-stitched designs are given for a family of epicycloids. (MP)

The free-energy change, or binding energy, of an idealized bubble cluster is calculated on the basis of one mole of gas, and on the basis of a single bubble going from sphere to polyhedron. Some new relations of bubble geometry are developed in the course of the calculation. (BB)

A method for converting a three-dimensional coordinate system to a two-dimensional coordinate system is explained. (MN)

A class of triangles related to the Pythagorean triangles is described; theorems concerning this class are proved. (SD)

All lines which bisect the area of a triangle envelope a three-cusped curve made up of three hyperbolic arcs. (MM)

The first of three articles showing how inductively-obtained results in transformation geometry may be organized into a deductive system. This article discusses two approaches to enlargement (dilatation), one using coordinates and the other using synthetic methods. (MM)

Worksheets are provided for use by students in grades 8 and above when sectioning a tetrahedron. Lesson objectives include the discovery of generalizations regarding the cross-sections of a tetrahedron. (MK)

Investigates the idea of the center of mass of a polygon and illustrates centroids of polygons. Connects physics, mathematics, and technology to produces results that serve to generalize the notion of centroid to polygons other than triangles. (KHR)

Presents a simple proof of the Pythagorean Theorem that only requires prior knowledge of elementary properties of triangles. (MKR)