# The Mathematical Underpinnings

HyPars™ toys and hyperbolic paraboloids fit best within the mathematical field of geometry. The hyperbolic paraboloid is one of only three surfaces that exist having the property of being “doubly ruled” – you can make the surface out of either of two independent sets of straight lines! The other two doubly ruled surfaces are the plane and the hyperboloid.

**Frequently Occurring Angles and Shapes**; Arrangements of assemblies of HyPars™ will frequently contain elements that are at angles of 35.26⁰, the angle a cubic diagonal makes with a cube face, 54.74⁰, the angle the cubic diagonals make with each other and 70.53⁰, the acute angle of a √2 rhombus.

Reoccurring shapes include hexagonal, square and lobed columns as well as rhombic and 4-HyPar shapes.

**Space Filling**; The L1 hyperbolic paraboloid (it has four equal length edges, designated L1, that are also the four equal length edges of a space filling tetrahedron) has some surprising mathematical properties. The space filling tetrahedron has four isosceles triangular faces with the triangles composed of two edges of length L1 and one edge of length L2 and the ratio of L1/L2 = 0.866. The L1 hyperbolic paraboloid has four equal length edges of length L1 and the distance between opposite corners is of length L2 – it is correlated with the space filling tetrahedron which has four connected edges of length L1 and two opposite non-connected edges of length L2.

Four of these L1 hyperbolic paraboloids can be arranged around a central axis to create what is called a 4-HyPar. These 4-HyPars can then be stacked together continuously in such a way that they completely fill space with no unenclosed volume and no overlaps – just like cubes could be stacked to completely fill space. It is this space filling characteristic that allows the HyPars™ to make so many different geometrical structures – there is no theoretical limit to the number of different structures that can be built.

**Edges Congruent with a Rhombohedron and Rhombic Dodecahedron**; Some unique assemblies of the L1 HyPars™ include six of them joined together to form a HyPar rhombohedron (with six √(2 ) (long to short axis ratio) rhombi) and 12 of them joined to form a HyPar rhombic dodecahedron. The edges formed by the L1 HyPar cells of these two assemblies are exactly congruent with the edges of the geometric “solids”, a rhombohedron and the rhombic dodecahedron. These rhombohedron and rhombic dodecahedron assemblies can also be joined to themselves continuously to fill space. Incidentally, the HyPar rhombic dodecahedron can be “tunneled”, i.e., it can be built with half the number of HyPars™ and the result is a rhombic dodecahedron with six tunnels through it! The common name for this assembly is the “Bloom” (That will make sense after assembling a few. They can even be made into a bouquet!) The HyPars™ Rhombohedron can also be made with half the number of parts resulting in left handed or right handed trefoils which can be used to create additional structures – e.g. a rhombohedron frame can be constructed out of HyPar Rhombohedron or the HyPar Trefoils.