One system, 60 ways to remove its kernel, and 10.5× between them

At its defaults it draws one system, 60 ways to remove its kernel, and 10.5× between them. A Laplacian is singular, so a solve has to remove its kernel, and the usual way is to delete one row and column — grounding a vertex, in the electrical reading. Which vertex is a free parameter that no account of the method mentions, and it is set by whichever index the code happens to drop. Every point here is one choice on the preferential 60: the condition number of the resulting positive definite matrix against the degree of the vertex removed. Grounding vertex 53, of degree 2, gives κ = 926.22; grounding vertex 3, of degree 17, gives 88.226 — a factor of 10.5 between two ways of solving the identical problem. The trend is that a high-degree vertex is the better ground, which has a reading: grounding a vertex fixes its potential, and fixing the potential of something the rest of the graph is strongly attached to constrains more of the graph.

grounding-choice is one function in lib/figures/graphlap.js — the matrix a graph makes — row sums that are exactly zero, a count that is a threshold, and a partition with no vector. Everything below came out of it during this build, at arguments taken from the essays rather than invented for this page. A figure here is the figure a reader meets in an essay, and if the generator changes, this page changes with it.

A Laplacian is singular, so a solve has to remove its kernel, and the usual way is to delete one row and column — grounding a vertex, in the electrical reading. Which vertex is a free parameter that no account of the method mentions, and it is set by whichever index the code happens to drop. Every point here is one choice on the preferential 60: the condition number of the resulting positive definite matrix against the degree of the vertex removed. Grounding vertex 53, of degree 2, gives κ = 926.22; grounding vertex 3, of degree 17, gives 88.226 — a factor of 10.5 between two ways of solving the identical problem. The trend is that a high-degree vertex is the better ground, which has a reading: grounding a vertex fixes its potential, and fixing the potential of something the rest of the graph is strongly attached to constrains more of the graph.

A Laplacian is singular, so a solve has to remove its kernel, and the usual way is to delete one row and column — grounding a vertex, in the electrical reading. Which vertex is a free parameter that no account of the method mentions, and it is set by whichever index the code happens to drop. Every point here is one choice on the star 30: the condition number of the resulting positive definite matrix against the degree of the vertex removed. Grounding vertex 1, of degree 1, gives κ = 898; grounding vertex 0, of degree 29, gives 1 — a factor of 898 between two ways of solving the identical problem. The trend is that a high-degree vertex is the better ground, which has a reading: grounding a vertex fixes its potential, and fixing the potential of something the rest of the graph is strongly attached to constrains more of the graph.

A Laplacian is singular, so a solve has to remove its kernel, and the usual way is to delete one row and column — grounding a vertex, in the electrical reading. Which vertex is a free parameter that no account of the method mentions, and it is set by whichever index the code happens to drop. Every point here is one choice on the preferential 60: the condition number of the resulting positive definite matrix against the degree of the vertex removed. Grounding vertex 53, of degree 2, gives κ = 926.22; grounding vertex 3, of degree 17, gives 88.226 — a factor of 10.5 between two ways of solving the identical problem. The trend is that a high-degree vertex is the better ground, which has a reading: grounding a vertex fixes its potential, and fixing the potential of something the rest of the graph is strongly attached to constrains more of the graph.

A Laplacian is singular, so a solve has to remove its kernel, and the usual way is to delete one row and column — grounding a vertex, in the electrical reading. Which vertex is a free parameter that no account of the method mentions, and it is set by whichever index the code happens to drop. Every point here is one choice on the grid 8×8: the condition number of the resulting positive definite matrix against the degree of the vertex removed. Grounding vertex 7, of degree 2, gives κ = 589.89; grounding vertex 28, of degree 4, gives 195.26 — a factor of 3.021 between two ways of solving the identical problem. The trend is that a high-degree vertex is the better ground, which has a reading: grounding a vertex fixes its potential, and fixing the potential of something the rest of the graph is strongly attached to constrains more of the graph.

A Laplacian is singular, so a solve has to remove its kernel, and the usual way is to delete one row and column — grounding a vertex, in the electrical reading. Which vertex is a free parameter that no account of the method mentions, and it is set by whichever index the code happens to drop. Every point here is one choice on the path 60: the condition number of the resulting positive definite matrix against the degree of the vertex removed. Grounding vertex 59, of degree 1, gives κ = 5735.6; grounding vertex 30, of degree 2, gives 1504.4 — a factor of 3.813 between two ways of solving the identical problem. The trend is that a high-degree vertex is the better ground, which has a reading: grounding a vertex fixes its potential, and fixing the potential of something the rest of the graph is strongly attached to constrains more of the graph.

A Laplacian is singular, so a solve has to remove its kernel, and the usual way is to delete one row and column — grounding a vertex, in the electrical reading. Which vertex is a free parameter that no account of the method mentions, and it is set by whichever index the code happens to drop. Every point here is one choice on the barbell 30: the condition number of the resulting positive definite matrix against the degree of the vertex removed. Grounding vertex 31, of degree 29, gives κ = 1053.9; grounding vertex 30, of degree 30, gives 959 — a factor of 1.099 between two ways of solving the identical problem. The trend is that a high-degree vertex is the better ground, which has a reading: grounding a vertex fixes its potential, and fixing the potential of something the rest of the graph is strongly attached to constrains more of the graph.

Every figure above checked its own claims on the way to being drawn, and a claim that failed would have stopped the picture rather than shipped a wrong one. Those checks used to leave no trace at all: a passing one returned true and the only evidence the figure had checked anything was that nothing crashed. The list below is what they actually said, collected by running this generator with an observer installed — not a description of what it is believed to check.

9 distinct claims across 6 sets of arguments, grouped below by shape — because most of them are one sentence with a different number in it, and how many separate times that sentence was put to the test is the informative part.

The rule does not apply to it. It factorises nothing, so there is no residual it could be withholding. That is worth stating rather than leaving blank: a site that reported the rule as satisfied by every generator would be counting mostly generators the rule never reached.

Across the library: the rule bites on 217 of 397 generators — 199 print a residual and 18 are exempt with a published reason; 180 factorise nothing. Read from lib/residual-rule.js, which is the same body the gate enforces from, and the gate's last check fails the build if this page and it disagree about any generator.