Machine Kinematics
solved, then drawn
Concept

Shear — where it appears

A deformation that slides parallel layers over one another, turning a square into a parallelogram while every length along the layers is kept. A pin-jointed grid's only motions are shears of whole columns and whole rows, which is why bracing one cell constrains a column and a row together.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

A network with no boundary at all. The kagome lattice, drawn out to 5 cells across and continuing for ever. The measurement is made on one cell: 3 joints, 6 bars, and a bar that leaves the cell comes back into it, written against the far end's position in the neighbouring cell. There is no boundary anywhere in the arithmetic, and the size of the network has gone from being a parameter to not existing. The highlighted triangle is the cell; every other line on the page is a copy of one of its six bars. positioned by solving, not by drawing.

The cell that repeats for ever

Take the size of a network to infinity and it stops being a parameter. What is left is one cell, six bars, and a question nobody has to ask about a finite assembly: does the pattern's period count as a body? A square grid is rigid if it does not and shears if it does, and so does the kagome.

networks · Network 5 braces, and it is rigid. A 3×3 grid of squares with 5 of its cells braced by a diagonal, and no freedom left. The bipartite graph on the 3 columns and 3 rows, with one edge per braced cell, has 1 component — and the number of freedoms is one less than that, at every bracing there is. Nothing in the rank computation knows about columns, rows or graphs.

Which diagonal rigidifies a grid

A three-by-three grid of squares needs five diagonals and eighty-one of the hundred and twenty-six ways of placing five will do. Which ones is not a rank question at all: it is whether a graph on the grid's columns and rows is connected, and eighty-one is the number of that graph's spanning trees.

networks · Network

Named alongside it

The objects these essays reach for when they reach for this one.

MobilityNetworkRankRedundant constraintDesign ruleEnumerationGrübler's criterionJoint graphKagomeLatticePeriodic frameworkSpanning tree

All concepts