Machine Kinematics
solved, then drawn
Concept

Kagome — where it appears

A planar lattice of corner-sharing triangles, three joints and six bars to a cell, whose count of freedoms is exactly nought per cell. That borderline makes it rigid or floppy on a knife-edge: a finite patch has one mechanism per missing bar, and turning its triangles decides whether those reach the middle.

Named by 4 essays across one field — each of them below, with the objects they name alongside it.

Also named here as lattice — the same set of essays touches all of them, so they are one junction rather than several.

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 The same patch twisted 17.2°: 35 mechanisms, all at the edge. A rhombus of 8 × 8 kagome cells — 192 joints, 346 bars — with every up-pointing triangle turned by 17.2° about its own centre. Each joint is drawn with an area proportional to its weight: its share of the diagonal of the projector onto the patch's mechanisms, which does not depend on how the mechanisms are written down and adds up over the joints to the number of mechanisms, 35. That number is Maxwell's count, 2 × 192 − 346 − 3, and the rank agrees with no redundant bar. The mean weight is 0.341 on the outermost ring of cells and 0.015 on the innermost. The twist kinks every line of bars at every joint, and the mechanisms fall away from the edge 23-fold in 3 cells.

The count says how many and not where

A kagome lattice has three joints and six bars in every cell and counts to exactly nothing, so a patch cut from it has as many mechanisms as its edge has lost bars: 5L − 5 for a rhombus of L cells a side, which the rank confirms at every size with no bar redundant. Straight or twisted, the number is the same. Where the mechanisms are is not: a straight patch keeps nearly half its edge weight in the middle, and a patch whose triangles are turned by 17° keeps a twentieth.

networks · Network A strip of distorted kagome with its mechanisms drawn by weight: one edge, the other, or both. A strip of kagome lattice 10 cells wide, periodic along its length (6 of its 12 cells drawn), with each up-triangle's three joints moved along the triangle's sides by 0.1, 0.1 and −0.1. Each joint is drawn with an area proportional to its share of the strip's mechanisms, summed over every wavevector along the strip. The cut leaves each long edge the same number of missing bars, so the count says the two edges share the mechanisms equally. They do when the three moves have one sign. With the third reversed, the top half carries 19.99 of the 24 and the bottom 4.01. The bulk's windings, measured from the twisted kagome's, are (1, −1).

An edge that knows which side it is on

Cut a strip from a kagome lattice and each long edge loses the same bars, so the count says the two edges share the strip's mechanisms equally. Move each triangle's three joints by the same signed amount and they do. Reverse one move and every mechanism that is not a translation moves to one edge. The shift is a whole number per cell, set by the winding of one six-by-six determinant round the lattice's zone. It changes only where one family of bars lines up straight.

networks · Network A seam that holds two mechanisms. Ten rows of a strip of kagome lattice sixty cells wide, centred on the line where two differently distorted lattices meet. Below it every up-triangle's joints are moved along its sides by 0.3, 0.3, −0.3, polarization −1 in the component that counts along the seam; above it by −0.3, −0.3, 0.3, polarization +1. No bar is missing anywhere along the seam. Each joint is drawn with an area proportional to its share of the strip's two mechanisms at one wavevector. They sit on the seam and die away from it into both lattices, and the strip's outer edges, out of the picture, have none. The wavevector along the seam is 3/12 of a turn per cell.

A seam with no bar missing

Join two distorted kagome lattices of opposite polarization along a line. Every joint on the line keeps all four of its bars, so Maxwell's count says the line holds nothing. It holds two per wavevector: two mechanisms one way up, two states of self-stress the other way up, exactly the difference of the two polarizations. The self-stresses are tensions the lattice carries with no load on it, along a line where no bar is spare. In a strip sixty cells wide the prediction holds at every wavevector a quarter of the zone or more from its centre.

networks · Network

Named alongside it

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

LatticeMobilityInfinitesimal flexNull spaceNetworkRankRedundant constraintGrübler's criterionPeriodic frameworkRedundancyShearUnit cell

All concepts