Atanasov--Ranganathan configuration 7, generic in the core #
This is the seventh local picture of Atanasov--Ranganathan, Proposition 5.1
(fig:configurations-for-genus-5, the scope commented %Seventh):
a b a, b carry chips
| | u, v are chip free and joined by a banana
u == v |u a| = |v b| (the figure labels both `a`)
Two chip-free core vertices joined by two parallel slots, each carrying one
further slot -- an arm -- to a chip vertex, and the two arms have equal
length. Interpolate the same negative height min |u a| |v b| on both
centres. The banana then has zero rise and moves nothing, each arm consumes at
most its own chip, and a shortest arm delivers a chip to its centre. With
the two arms equal, one script therefore reaches both centres.
Why the rows need it, and why the arms are marked. AR's sixth and seventh
genus-five families (atlas rows 05 and 08) each contain two of these
pictures, and in each of them one of the two arms is half of a slot: the chip
sits at an interior point whose offset is a length, which is exactly how the
figure arranges for the two arms to be equal. So this file states the arm
ledger for a marked slot as well as an unmarked one, on top of
ConfigurationMarkedCommon. A marked arm is an ordinary arm of the half
length:
- centre at the tail, chip at the mark: the centre sees the canonical ramp
of rise
heightovermark esteps, and the slot's head end sees nothing; - centre at the head, chip at the mark: the centre sees the canonical ramp
of rise
-heightoverlength e - mark esteps, and the tail end sees nothing.
Both readings are ConfigurationCommon's one-slot ramp lemmas at the half
length, so nothing new is proved about ramps here.
Composability. Like ConfigurationTwo, the conclusions are stated one
centre at a time; a row declares which of its chip-free vertices form banana
pairs and covers the rest by other pictures. Rows 05 and 08 pair this file
with ConfigurationThree.
The shared height #
The height interpolated on both centres of a banana pair: the shorter of the two arm lengths. On the AR rows the two arms are equal, so both centres are targets of the same script.
Equations
- AtanasovRanganathan.ConfigurationSeven.bananaHeight first second = min first second
Instances For
The banana slots move nothing #
Both centres carry the same height, so each parallel slot has zero rise.
The two parallel slots of a banana whose ends carry the same potential contribute nothing anywhere.
A marked arm, read from its tail #
The centre is the tail of e, its chip sits at the mark, and the script is flat
beyond the mark: markRiseIn e = height, markRiseOut e = 0.
The rise data of an arm read from its tail.
What the centre receives along a marked arm read from its tail: the first
slope of the canonical ramp of rise height over mark e steps.
The far end of a marked arm read from its tail receives nothing: the script is already flat there.
A shortest marked arm delivers one chip to its centre.
A marked arm never takes a chip away from its centre.
A marked arm, read from its head #
The centre is the head of e, its chip sits at the mark, and the script is flat
before the mark: markRiseIn e = 0, markRiseOut e = -height.
The near end of a marked arm read from its head receives nothing.
What the centre receives along a marked arm read from its head: minus the
last slope of the canonical ramp of rise -height over length e - mark e
steps.
A shortest marked arm delivers one chip to its centre, read from the head.
A marked arm never takes a chip away from its centre, read from the head.