Structural rigidity in the context of Cycle graph

Solid is a state of matter in which atoms are closely packed and are difficult to move past each other. Solids resist compression, expansion, or external forces that would alter its shape, with the degree of resistance being dependent upon the specific material under consideration. Solids also always possess the least amount of kinetic energy per atom/molecule relative to other phases or, equivalently stated, solids are formed when matter in the liquid / gas phase is cooled below a certain temperature. This temperature is called the melting point of the substance and is an intrinsic property; i.e. independent of how much of the matter there is. The vast majority of substances, when in the solid state, can be arranged in one of a few ubiquitous structures.

Solids are characterized by structural rigidity and resistance to applied external forces and pressure. Unlike liquids, solids do not flow to take on the shape of their container, nor do they expand to fill the entire available volume like a gas. Much like the other three fundamental phases, solids also expand when heated, the thermal energy put into increasing the distance and reducing the potential energy between atoms. However, solids do this to a much lesser extent. When heated to their melting point or sublimation point, solids melt into a liquid or sublimate directly into a gas, respectively. For solids that directly sublimate into a gas, the melting point is replaced by the sublimation point. As a rule of thumb, melting will occur if the subjected pressure is higher than the substance's triple point pressure, and sublimation will occur otherwise. Melting and melting points refer exclusively to transitions between solids and liquids. Melting occurs across a great extent of temperatures, ranging from 0.10 K for helium-3 under 30 bars (3 MPa) of pressure, to around 4,100 K at 1 atm for the composite refractory material hafnium carbonitride.

Crystal growth is a major stage of a crystallization process, and consists of the addition of new atoms, ions, or polymer strings into the characteristic arrangement of the crystalline lattice. The growth typically follows an initial stage of either homogeneous or heterogeneous (surface catalyzed) nucleation, unless a "seed" crystal, purposely added to start the growth, was already present.

The action of crystal growth yields a crystalline solid whose atoms or molecules are close packed, with fixed positions in space relative to each other.The crystalline state of matter is characterized by a distinct structural rigidity and very high resistance to deformation (i.e. changes of shape and/or volume). Most crystalline solids have high values both of Young's modulus and of the shear modulus of elasticity. This contrasts with most liquids or fluids, which have a low shear modulus, and typically exhibit the capacity for macroscopic viscous flow.

Structural engineering is a sub-discipline of civil engineering in which structural engineers are trained to design the 'bones and joints' that create the form and shape of human-made structures. Structural engineers also must understand and calculate the stability, strength, rigidity and earthquake-susceptibility of built structures for buildings and nonbuilding structures. The structural designs are integrated with those of other designers such as architects and building services engineer and often supervise the construction of projects by contractors on site. They can also be involved in the design of machinery, medical equipment, and vehicles where structural integrity affects functioning and safety. See glossary of structural engineering.

Structural engineering theory is based upon applied physical laws and empirical knowledge of the structural performance of different materials and geometries. Structural engineering design uses a number of relatively simple structural concepts to build complex structural systems. Structural engineers are responsible for making creative and efficient use of funds, structural elements and materials to achieve these goals.

A geodesic dome is a hemispherical thin-shell structure (lattice-shell) based on a geodesic polyhedron. The rigid triangular elements of the dome distribute stress throughout the structure, making geodesic domes able to withstand very heavy loads for their size.

In geometry, a Schönhardt polyhedron is a polyhedron with the same combinatorial structure as a regular octahedron, but with dihedral angles that are non-convex along three disjoint edges. Because it has no interior diagonals, it cannot be triangulated into tetrahedra without adding new vertices. It has the fewest vertices of any polyhedron that cannot be triangulated. It is named after the German mathematician Erich Schönhardt, who described it in 1928, although the artist Karlis Johansons had exhibited a related structure in 1921.

One construction for the Schönhardt polyhedron starts with a triangular prism and twists the two equilateral triangle faces of the prism relative to each other, breaking each square face into two triangles separated by a non-convex edge. Some twist angles produce a jumping polyhedron whose two solid forms share the same face shapes. A 30° twist instead produces a shaky polyhedron, rigid but not infinitesimally rigid, whose edges form a tensegrity prism.

In physics, the line of action (also called line of application) of a force (F→) is a geometric representation of how the force is applied. It is the straight line through the point at which the force is applied, and is in the same direction as the vector F→. The lever arm is the perpendicular distance from the axis of rotation to the line of action.

The concept is essential, for instance, for understanding the net effect of multiple forces applied to a body. For example, if two forces of equal magnitude act upon a rigid body along the same line of action but in opposite directions, they cancel and have no net effect. But if, instead, their lines of action are not identical, but merely parallel, then their effect is to create a moment on the body, which tends to rotate it.