Many Pharaoh players, including the author, would love to know more about which way and how far roaming walkers presented with different road geometries will walk This article is for them.

After posting an earlier article on the walks of labor recruiters on road intersections, I firmly resolved to give the quadramble problem - and myself - a rest. Alas, when I rest, I play Pharaoh. At the conclusion of the labor recruiter study, I had a plethora of intriguing additional questions about roamer behavior, but two seemed paramount in importance:

1. To what extent could the strict conditions originally imposed on those experimental road-plus-building geometries that excluded all housing and other buildings from the vicinity of tested roamer-generating structures be relaxed without altering the quadrambles walked by roamers emitted by those buildings?

2. What are the size and shape of the area around buildings outside of which alterations in road geometry had little or no effect on the quadrambles of the buildings' roamers?

The answer to question 1 came quickly and is described in replies to my original post on labor recruiters (StephAmon 26DEC01, reply 13): Adding housing to a road+building problem does not change the quadramble of the labor recruiter emitted by the building. Instead, it changes the identity of the roamer who walks the quadramble. Close passage of two huts caused a labor recruiter's building to generate a service walker (if that building is the sort that ever does) but does not suppress further emission of labor recruiters. Passage of eight huts on each leg of a service walker's quadramble (usually - Magistrates may be more demanding.) ensured that his/her building would emit no additional labor recruiters beyond the first. Passage of three or four huts per quadramble leg caused the building to emit a labor recruiter and service walker nearly simultaneously, but with the recruiter always appearing a few animation frames before the service walker. In all cases, the recruiter and service walker from the same building shared the same quadramble. Moreover, they executed walks from the same queue of walks. If a building's quadramble said "Go NE, then SE, and SW, and finally NW" and the labor recruiter appeared and went NE, a few animation frames later the service walker would appear and head SE on the next leg of the quadramble. Thus, quadrambles appeared to be owned by buildings rather than their walkers, and further quadramble studies could be conducted by watching service walkers, which are more easily distinguished from those of other buildings than are labor recruiters.

The answer to question 2 did not yield itself to casual probing, and the study described here was conducted pursue it in a brute-force fashion - meaning I was forced to beat the question into submission by collecting so much data under controlled conditions that the game could not hide the signal I was looking for through the noise.

Originally, I wanted to record quadrambles from buildings located at increasing distances from corners: road structures that players use abundantly in their cities. The data that I gathered then might have had immediate applicability to actual play. Alas, the designers must have anticipated that we would use a lot of corners in our cities and worked hard to make the quadrambles of buildings near them vary in a fashion that was so smooth and elegantly continuous with distances increasing even out to 11 squares from the corner that I could not see the abrupt transitions in quadrambles at some kind of go-blind distance that I was seeking. Clearly, I was looking at the output of highly refined code. I needed a way to look into less carefully refined areas of the algorith. Reasoning that the designers would not expect players to make as many observations of walkers at four-way intersections as they would at corners and therefore would not have refined the algorithm as thoroughly to handle them, I tried backing buildings of various sizes further and further away from road X's and got the sharp, clean discontinuity I wanted along a SE road from a four-way intersection. Buildings located six squares away along the SE road (with the building on the SW side of the road) from a four-way intersection send their walkers on the same clockwise rotation of walks through all four roads of such an intersection that I described in my earlier post for (the labor recruiters from) buildings located immediately adjacent to road X's. Buildings located seven squares away from the intersection on the same side of the same road instead dispatched their walkers on pure straight-road quadrambles that were identical to the walk cycles of labor recruiters from buildings located on vast, perfectly straight roads in the desert - located dozens of squares and across the Nile from any other sprites, or road squares, or structures (except gardens). One could not wish for a louder, clearer announcement by the quadramble-generating algorithm that, in some way, it could not "see" the intersection when it generated the quadrambles for walkers from the latter buildings.

In this communication, I offer the quadrambles of 1x1, 2x2, and 3x3 service walker-emitting buildings at increasing distances along both sides of all four roads from a four-way intersection. At the end of the article, these data are used to deduce the unexpected (by me, at least) size and shape of the domain of strong road influence (or simply, "Domain") around each walker-emitting building within which the road geometry seems powerfully to dictate the quadrambles of the building's roamers. Nothing said here should imply that road geometries and other game variables, like the Gods (Cartouche Bee 12JAN02, in reply 6 to Martinu,02OCT00), outside of a buildings domain can exert no influence on quadramble generation. I merely intend to assert that most buildings, most of the time look primarily at the roads within their own domains (probably with a little support from invoked routines used to "peer" further down roads) when calculating the quadrambles on which to dispatch their walkers.

Materials and Methods

Software. All tests were conducted using Pharaoh with the Cleopatra Expansion Pack installed and with the Cleo patch applied.

Game conditions. Game difficulty was set to Normal for all studies described here. I ran the game at absolute top speed during most of the periods of observation of roamer quadrambles, except when the walkers neared their turn-around squares, when speed was swiftly adjusted to about 20%. Strictly speaking, these should be called "runner studies" rather than "walker studies", but the sprites seemed to follow the same quadrambles whether they were creeping, walking, or racing.

Maps. The quadrambles described in this report were observed for service walkers and labor recruiters in Cleopatra's Alexandria. Selected quadrambles that were interpolated from actual data were spot checked in Rostja and Men-nefer as noted in the tables to verify their reliability as well as to explore their generalizability to other maps of the results reported here.

Layout. In all maps, a huge X made of four roads intersecting at a four-way intersection was constructed so that every road from the intersection ran straight for at least 26 squares from the intersection and extended for a total travel distance of at least 60 squares from the intersection. Eight huts were distributed along each of the four roads in Cleo's Alexandria to suppress the appearance of all but the initial labor recruiter from tested buildings (except for Courthouses which needed more huts). Architects and firehouses were installed to protect the test buildings during long repeated iterations of quadrambles suspected of instability.

I freely alternated between almost all the kinds of 1x1 through 3x3 buildings that dispatch roaming service walkers including libraries but not mortuaries. Cleopatra's Alexandria can import just about anything one could want by land except linen. I did not include industrial or other buildings that dispatch labor recruiters as their only walkers in this study.

Terminology. Because the roads from a four-way intersection divide the map into quadrants in much the same way as the X and Y axes divide a Cartesian plane, I refer to buildings lying north of both roads a being in the north quadrant and buildings east of both roads as being in the east quadrant, etc.

I follow Brugle's practice in treating North as lying to the upper left in glyphy. Thus, a horizontal road in glyphy would correspond to a NW-to-SE road on the screen. I could have represented both on-road locations and off-road locations using Cartesian coordinates centered on an origin, square (0, 0), at the four-way intersection of the Y with the X axis corresponding to the horizontal road in glyphy. This would have given a clear, but not terribly compact method for representing locations, but particularly for on-road squares it seemed verbose. I prefer the tidier notation that results from imposing a complex coordinate system on the intersection in which the real axis is the NW-to-SE road and the imaginary axis is the NE-to-SW road. To record the location of a road square (perhaps one where a walker turned around) that was 34 squares NE of the intersection would require writing (0, 34) in Cartesian coordinates, but simply 34i in complex notation. Addition and subtraction for complex coordinates work just like they do for Cartesian ones: (a, b) + (c, d) = (a + c, b + d) and a+bi + c+di = (a+c)+(b+d)i. The real and imaginary components of the number are added (or subtracted) separately. In the present work, no multiplication of coordinates was required, so the rules for multiplying complex numbers have no relevance here.

This work also requires a succinct way of expressing the locations of buildings as they were moved further and further away from the four-way intersection on both sides along all four roads. The most compact notation I found was to specify a building's location by recording the quadrant in which it lay and an offset specifying the direction and number of squares it had been moved away from the intersection. A building immediately adjacent to a four-way intersection on its north side would be in north quadrant at an offset of zero. If such a building were moved further and further to the northeast, it would pass through offsets i, 2i, 3i, 4i, etc. If the building instead were moved further and further along the road to the NW, it would lie at offsets -1, -2, -3, etc. I tried recording a building's location by recording the complex coordinate of its north square but became too frequently confused for buildings larger than 1x1 to want to use that method. Besides, the notation SQ-5i is nice quick way of specifying that a building lies on the south side (in the south quadrant) of the road to the southwest (the -i road) and has five squares between it and the intersection. Admittedly, a small potential source of confusion remains, because the road square touched by such a building that lies closest to the intersection is actually at -6i.

Results

Table 1 presents the turn-around squares observed for service-providing short walkers from 1x1 buildings located at various offsets from a four-way intersection. As was seen for labor recruiters from buildings on four-way intersections, service walkers near a four-way intersection bracketed target points near their buildings of origin by making walks of lengths sufficient to take them 34 squares NE, SE, SW, and NW (in that clockwise order, although they can start in any of the four directions) of the intersection, if the roads had been straight. I have yet to find a map in the game in which I could make all four roads straight for 60 squares, but by the time they had gone 26 squares away from the intersection the walkers only seemed to care about the maximum travel distance specified in their orders and walked that far. They did not walk far enough to get to absolute distances of 34 squares in a particular compass direction from their target points - only far enough to have gotten there if the roads had been perfectly straight.

Table 1a. Quadrambles of short walkers from 1x1 buildings and the squares targeted by bracketing walks within the quadrambles at different offsets from a four way intersection. The numbers shown are the locations of the squares on which the roamers turned around and reversed course on the four legs of their quadrambles in complex notation, as described in the methods.

Road to Southeast
Offset	South quadrant		East quadrant
	Quadramble	Target	Quadramble	Target
0	[34, -34i, -34, 34i]	0	[34, -32i, -34, 36i]	0+2i
1	[35, -33i, -33, 35i]	1+i	[35. -31i, -33, 37i]	1+3i
2	[36, -32i, -32, 36i]a	2+2i	[36, -30i, -32, 38i]	2+4i
3	[37, -31i, -31, 37i]a	3+3i	[37, -29i, -31, 39i]a	3+5i
4	[38, -30i, -30, 38i]a	4+4i	[38, -28i, -30, 40i]aR	4+6i
5	[39, -29i, -29, 39i]	5+5i	[39, -27i, -29, 41i]	5+7i
6	[40, 33, -28, 33]	srq(6)b	[40, 33, 28i, 33]	msrq(6)b
Road to Southwest
Offset	West quadrant		South quadrant
	Quadramble	Target	Quadramble	Target
0	[32, -34i, -36, 34i]	-2	[34, -34i, -34, 34i]	0
-i	[31, -35i, -37, 33i]	-3-i	[33, -35i, -35, 33i]	-1-i
-2i	[30, -36i, -38, 32i]a	-4-2i	[32, -36i, -36, 32i]a	-2-2i
-3i	[29, -37i, -39, 31i]a	-5-3i	[31, -37i, -37, 31i]a	-3-3i
-4i	[28, -38i, -40, 30i]aM	-6-4i	[30, -38i, -38, 30i]a	-4-4i
-5i	[27, -39i, -41, 29i]	-7-5i	[29, -39i, -39, 29i]	-5-5i
-6i	[-19, -40i, -19, 28i]	msrq(-6i)	[19, -40i, -19, 28i]	msrq(-6i)

Table 1b. Quadrambles of short walkers from 1x1 buildings and the squares targeted by bracketing walks within the quadrambles at different offsets from a four way intersection.

Road to Northwest
Offset	West quadrant		North quadrant
	Quadramble	Target	Quadramble	Target
0	[32, -34i, -36, 34i]	-2	[32, -32i, -36, 36i]	-2+2i
-1	[31, -33i, -37, 35i]b	-3+i	[31, -31i, -37, 37i]	-3+3i
-2	[30, -32i, -38, 36i]a	-4+2i	[30, -30i, -38, 38i]a	-4+4i
-3	[29, -31i, -39, 37i]a	-5+3i	[29, -29i, -39, 39i]aM	-5+5i
-4	[28, -30i, -40, 38i]aM	-6+4i	[28, -28i, -40, 40i]aR	-6+6i
-5	[27, -29i, -41, 39i]	-7+5i	[27, -27i, -41, 41i]	-7+7i
-6	[26, 19i, -42, 19i]	msrq(-8)	[28i, 19i, -42, 19i]	?
-7	NDc		[28i, 18i, -43, 18i]	?
-8	ND		[28i, 17, -44, 17]	?
-9	ND		[23, 16, -45, 16]	srq(-11)
Road to Northeast
Offset	North quadrant		East quadrant
	Quadramble	Target	Quadramble	Target
0	[32, -32i, -36, 36i]	-2+2i	[34, -32, -34, 36i]	+2i
i	[31, -31i, -37, 37i]	-3+3i	[33, -31i, -35, 37i]	-1+3i
2i	[30, -30i, -38, 38i]a	-4+4i	[32, -30i. -36, 38i]a	-2+4i
3i	[29, -29i, -39, 39i]a	-5+5i	[31, -29i, -37, 39i]a	-3+5i
4i	[28, -28i, -40, 40i]a	-6+6i	[30, -28i, -38, 40i]aM	-4+6i
5i	[27, -27i, -41, 41i]	-7+7i	[29, -27i, -39, 41i]	-5+7i
6i	[33i, -28, 33i, 42i]	?	[33i, 26, 33i, 42i]	msrq(8i)
7i	[34i, 25, 34i, 43i]	msrq(9i)	ND

^aQuadramble was interpolated using data from Cleopatra's Alexandria.
^bStraight road quadramble or modified straight road quadramble bracketing the point specified in parentheses in only one direction.
^cNot determined.
^MQuadramble was corroborated by observation of roamers in Men-Nefer.
^RQuadramble was corroborated by observation of roamers in Rostja.

Bracketing and targets. The target squares bracketed by the walks of service providers from buildings near an intersection can be calculated by averaging the NW and SE turn-around locations of the walker to get the "real" component of the target and by averaging the NE and SW turn-around location to get the target's imaginary component. For example, a dentist or apothecary moved five squares away from a four-way intersection along the SW side of the road to the NW (Northwest road, offset -5 in Table 1) executes a repeating cycle of four walks (quadramble) that has him turning around on the 27th road square SE of the intersection, square 29 SW of the intersection, 41 squares NW of the intersection, and on the 39^th square NE of the intersection. The real component of his target square is (27 - 41)/2 = -7, and the imaginary component is (39i - 29i)/2 = 5i. Thus, his walks "bracket" the square -7+5i, as shown in Table 1. The turn-around squares in the walks of architects, firemen, and police officers from buildings at the offsets shown in the table are identical to those of the 1x1 short walkers, except that they are all extended by 17 squares, which seems to be a kind of long-walker extension factor. When averages are taken, however, the calculated target squares are the same for long walkers as for short walkers. Policemen, like other "inflexible bureaucrats" (Brugle 09DEC00) use walk-starting squares different from those of common walkers. Nevertheless, near an X intersection, their turn-around squares are the same as those of firemen and architects. Clearly, the algorithm calculates how many squares a policeman must travel in each direction to reach the desired bracketing square in a way that corrects for the difference in his walk-starting square.

The locations of the turn-around squares for walkers from near X intersections and of the squares their walks targeted formed strikingly regular patterns as functions of distance from the intersection (Table 1). In the east quadrant of the intersection, for example, buildings located 1 square (at offset 0) or "right next to" the intersection sent their (short) walkers on quadrambles that had them turning around on square 36i on the road to the northeast. Nearby buildings located 2, 3, or 4 squares from the same intersection (at offsets 1, 2, and 3, respectively on the road to the southeast) launched walkers on cycles that took them to squares 37i, 38i, and 39i on the northeast legs of their quadrambles. I relied on the regularity of these patterns to calculate many quadrambles by interpolating the values in actually observed data. Naturally, I did not fully trust such interpolated data, so I spot checked them in other cities as described below and as noted using upper-case footnotes in the tables.

The positions of building targets about the intersection changed as a function of the building offsets in adherence to a simple rule: for each square the building is offset from the intersection, the target point moves along with the building, but it also moves one square NE if the building is on a real (NW2SE) road or one square NW if the building is on an imaginary (NE2SW) road. The offset may have either sign. Fig. 1 diagrams the targeted squares for a few series of 1x1 buildings to show this rule in operation.

Figure 1. Series of buildings in each of the quadrants of a four-way intersection and the squares targeted by the bracketing walks of service walkers from those buildings. North lies towards upper left. Firemen from each of the three firehouses target the plaza square touching their own building's north corner. Architects bracket the shrines, and the policemen apothecaries and dentists target squares marked by huts, wells, and small statues, respectively.

Loss of vision. A 1x1 building in the south quadrant of a four-way intersection located at an offset of 5 along the road to the southeast clearly "sees" the four way intersection, as shown by the fact that the walker it dispatches will travel down all four roads in clockwise rotating succession (Table 1). If the building is moved just one square further out along the road to offset 6, the building (in generating its quadrambles) no longer acts as if it "sees" the target. Instead, it dispatches walkers on a long-straight-road quadramble (Table 1). For purposes of comparison, the straight road quadramble of short service walkers are the same as those of labor recruiters given in reply 4 Table 4 in StephAmon (26DEC01). The two longer walks, 40 and -28, bracket a real value of 6, which happens to be the road square touching the building's north corner. Thus, the target square is no longer positioned with any imaginary-axis reference to the building's distance from the intersection. The walks that turn around on road square 33 are short default walks of 26 squares out the door, since common walkers from this building use road square 7 as their walk-starting point. Across the road in the east quadrant an equally sharp discontinuity occurs between lags of 5 and 6 from the intersection. Unlike the case in the south quadrant, in the east quadrant at a lag of 6 the walkers are dispatched on a quadramble the looks like some kind of a modification of the usual straight-road quadramble for a 1x1 building NE of a NW2SE road, in which the leg that usually goes to -28 took a turn at the intersection and bent the walker 28 squares to the northeast at 28i instead.

Along the road to the southwest, 1x1 buildings also lost site of the intersection at an offset of -6i. On neither side of the street (south quadrant or west quadrant) did walkers from such buildings execute a perfect straight road quadramble. In two of these cases, the bracketing walks were in straight-road orientation, but the default short walks took wrong turns. Walkers from the 1x1 on the south side of the road actually used all four roads from the intersection. However, we can clearly identify the walks that turned around on road squares 18 and -18 as short placeholders by their lengths, since they require 26 squares of travel to reach from the building's walk-starting square at -7i. Walkers from buildings in the east quadrant along a NE road or in the west quadrant along a NW road also showed a sharp transition in the quadrambles between absolute lags of 5 and 6 from the intersection. At lags of -6 and 6i, these buildings dispatched their walkers on some kind of directionally modified straight-road quadramble.

The quadrambles of walkers from 1x1 buildings in the north quadrant of a four-way intersection also displayed sharp discontinuities between lags of -5 and -6 (along the NW road) and between 5i and 6i (along the road to the NE). At -6 and 6i, neither building could "see" the roads to the southeast and southwest, but neither building completely changed its walker's quadrambles to a recognizable modification of straight-road form until greater distances from the intersection were reached at lags of -9 and 7i. Nevertheless, a recognizable "loss of vision" by buildings located seven squares from the intersection occurred for north-quadrant 1x1 buildings just as it had for 1x1 buildings in the other three quadrants. Thus, the area around a 1x1 building that is most minutely examined by the quadramble-generating algorithm seems to extend symmetrically in all four directions for six squares.

Asymmetry of vision. For 2x2 and 3x3 buildings, the distance at which the quadramble-generating algorithm appeared to lose sight of the intersection varied depending upon in which compass direction the intersection lay from the building. A 2x2 building in the south quadrant of a four-way intersection could "see" just as far as its 1x1 counterpart A south-quadrant 2x2 building switched from a quadramble that fully exploited (with bracketing walks) all four directions available from the intersection at an offset of 5 to a pure straight road quadramble at an offset of 6, just like a 1x1 building would do (Table 2). In the east and west quadrant, buildings lost site of the intersection (as shown by abrupt discontinuities in the quadrambles at increasing offsets) at offsets of 5, 5i, -5, or -5i (Table 2). In the north quadrant, the quadrambles of service walkers from 2x2 buildings displayed their first discontinuities when the buildings were moved from an offset of -3 to -4 along the road to the NW and when the building on the NE road was moved from an offset of 3i to 4i.

Table 2a. Quadrambles recorded as turn-around squares of short walkers from 2x2 buildings at different offsets from a four way intersection and the square targeted by bracketing walks within each quadramble.

Road to Southeast
Offset	South quadrant		East quadrant
	Quadramble	Target	Quadramble	Target
0	[34, -34i, -34, 34i]	0	[33, -31i, -35, 37i]	-1+3i
1	[35, -33i, -33, 35i]	1+i	[34, -30i, -34, 38i]	0+4i
2	[36, -32i, -32, 36i]aM	2+2i	[35, -29i, -33, 39i]	1+5i
3	[37, -31i, -31, 37i]a	3+3i	[36, -28i, -32, 40i]	2+6i
4	[38, -30i, -30, 38i]M	4+4i	[37, -27i, -31, 41i]	3+7i
5	[39, -29i, -29, 39i]	5+5i	[38, +26i, 30i, 42i]	(4+8i)
6	[40, 33, -28, 33]	srq(6)	[39, 25i, 29i, 34]	(5)+?i
7	ND		[40, -24i, 29i, 35]	?
8	ND		[41, -23i, 29i, 36]	?
9	ND		[42, -22i, 30i, 37]	?
10	ND		[43, -21i, -25 | 25i, 38]	msrq(9)
Road to Southwest
Offset	West quadrant		South quadrant
	Quadramble	Target	Quadramble	Target
0	[31, -33i, -37, 35i]	-3+i	[34, -34i, -34, 34i]	0
-i	[30, -34i, -38, 34i]aR	-4	[33, -35i, -35, 33i]a	-1-i
-2i	[29, -35i, -39, 33i]aM	-5-i	[32, -36i, -36, 32i]aM	-2-2i
-3i	[28, -36i, -40, 32i]a	-6-2i	[31, -37i, -37, 31i]a	-3-3i
-4i	[27, -37i, -41, 31i]	-7-3i	[30, -38i, -38, 30i]	-4-4i
-5i	[26i, -38i, -42, 30i]	(-8)-4i	[29, -39i, -39, 29i]	-5-5i
-6i	[25i, -39i, -20, -29]	?+(-5i)	[18 | 18i, -40i, 18i, 28i]	msrq(-6i)
-7i	[24, -40i, 18i, -29]b	?	[17, -41i, 17i, 27i]d	msrq(-7i)
-8i	[23, -41i, 17i, -29]	?	ND
-9i	[22, -42i, ,16, -30]	?	ND
-10i	[-21, -43i, 15, 25i]	msrq(-9i)	ND

Table 2b. Quadrambles of short walkers from 2x2 buildings and the squares targeted by quadramble bracketing walks at different offsets from a four way intersection.

Road to Northwest
Offset	West quadrant		North quadrant
	Quadramble	Target	Quadramble	Target
0	[31, -33i, -37, 35i]	-3+i	[30, -30i, -38, 38i]	-4+4i
-1	[30, -32i, -38, 36i]a	-4+2i	[29, -29i, -39, 39i]	-5+5i
-2	[29, -31i, -39, 37i]aM	-5+3i	[28, -28i, -40, 40i]a	-6+6i
-3	[28, -30i, -40, 38i]aR	-6+4i	[27, -27i, -41, 41i]	-7+7i
-4	[27, -29i, -41, 39i]	-7+5i	[30i, -38, -42, 42i]b	?
-5	[26, 19i, -42, 19i]	msrq(-8)	[29i, -39, -43, 20i]b	?
-6	ND		[29i, -40, -44, 19i]b	?
-7	ND		[29i, -41, -45, 18i]	?
-8	ND		[30i, -42, -46, 17i]	?
-9	ND		[21 | -21i, -43,-47, 16]	msrq(-13)
Road to Northeast
Offset	North quadrant		East quadrant
	Quadramble	Target	Quadramble	Target
0	[30, -30i, -38, 38i]	-4+4i	[33, -31i, -35, 37i]	-1+3i
i	[29, -29i, -39d, 39i]	-5+5i	[32, -30i, -36, 38i]	-2+4i
2i	[28, -28i, -40, 40i]a	-6+6i	[31, -29i, -37, 39i]a	-3+5i
3i	[27, -27i, -41, 41i]M	-7+7i	[30, -28i, -38, 40i]	-4+6i
4i	[38i, -30, -42, 42i]	?	[29, -27i, -39 ,41i]	-5+7i
5i	[39i, -29, 33i, 43i]	?	[32i, 26, 32i, 42i]	msrq(8i)
6i	[40i, -29, 34i, 44i]	?	ND
7i	[41i, -29, 35i, 45i]	?	ND
8i	[42i, -22i, 36i, 46i]	srq(12i)	ND

^aInterpolated data. Other footnotes as in Table 1.

(Continues)

Table 3a. Quadrambles of short walkers from 3x3 buildings at different offsets from a four way intersection and the square targeted by bracketing walks within each quadramble.

Road to Southeast
Offset	South quadrant		East quadrant
	Quadramble	Target	Quadramble	Target
0	[34, -34i, -34, 34i]	0	[32, -30i, -36, 38i]	-2+4i
1	[35, -33i, -33, 35i]	1+i	[33, -29i, -35, 39i]a	-1+5i
2	[36, -32i, -32, 36i]	2+2i	[34, -28i, -34, 40i]a	6i
3	[37, -31i, -31, 37i]a	3+3i	[35, -27i, -33, 41i]	1+7i
4	[38, -30i, -30, 38i]aM	4+4i	[36, 26i, 32i, 42i]	(2+8i)
5	[39, -29i, -29, 39i]	5+5i	[37, 25i, 31i, 43i]	(3+9i)
6	[40, 33, -28, 33]	srq(6)	[38, -24i, 30i, 35]	msrq(4)c
Road to Southwest
Offset	West quadrant		South quadrant
	Quadramble	Target	Quadramble	Target
0	[30, -32i, -38, 36i]	-4+2i	[34, -34i, -34, 34i]	0
-i	[29, -33i, -39, 35i]a	-5+i	[33, -35i, -35, 33i]a	-1-i
-2i	[28, -34i, -40, 34i]a	-6	[32, -36i, -36, 32i]a	-2-2i
-3i	[27, -35i, -41, 33i]	-7-i	[31, -37i, -37, 31i]a	-3-3i
-4i	[26i, -36i, -42, 32i]	(-8-2i)	[30, -38i, -38, 30i]aR	-4-4i
-5i	[25i, -37i, -43, -31]	(-9-3i)	[29, -39i, -39, 29i]	-5-5i
-6i	[24, -38i, -19, -30]	msrq(-4i)	[17i | 17, -40i, 17i, 28i]	srq(-6i)

Table 3b. Quadrambles of short walkers from 3x3 buildings and the squares targeted by quadramble bracketing walks at different offsets from a four way intersection.

Road to Northwest
Offset	West quadrant		North quadrant
	Quadramble	Target	Quadramble	Target
0	[30, -32i, -38, 36i]	-4+2	[28, -28i, -40, 40i]	-6+6i
-1	[29, -31i, -39, 37i]a	-5+3	[27, -27i, -41, 41i]	-7+7i
-2	[28, -30i, -40, 38i]a	-6+4	[32i, -36, -42, 42i]	?
-3	[27, -29i, -41, 39i]	-7+5	[31i, -37, -43, 43i]	?
-4	[26i, 19i, -42, 19i]	msrq(-8)	[30i, -38, -44, -21i]	?
-5	ND		[30i, -39, -45, 20i]	?
-6	ND		[30i, -40, -46, 19i]	?
-7	ND		[31i, -41, -47, 18i]	?
-8	ND		[32i, -42, -48, 17i]	?
-9	ND		[19i, -43, -49, 16]	msrq(-15)
Road to Northeast
Offset	North quadrant		East quadrant
	Quadramble	Target	Quadramble	Target
0	[28, -28i, -40, 40i ]	-6+6i	[32, -30i, -36, 38i]	-2+4i
i	[27, -27i, -41,41i]	-7+7i	[31, -29i, -37, 39i]aM	-3+5i
2i	[36i, -32, -42, 42i]	?	[30, -28i, -38, 40i]aR	-4+6i
3i	[37i, -31, -43, 43i]	?	[29, -27i, -39, 41i]	-5+7i
4i	[38i, -30, 33i, 44i]	?	[31i, 26, 31i, 42i]	msrq(8i)
5i	[39i, -30, 34i, 45i]	?	ND
6i	[40i, -30, 35i, 46i]	?	ND
7i	[41i, -31, 36i, 47i]	?	ND
8i	[42i, -20, 37i, 48i]	msrq(14i)	ND

^aInterpolated data. Other footnotes as in Table 1.

The "go-blind" positions at which 2x2 buildings seemed first to lose clear site of the intersection are shown on the left side of Fig. 2. As was done for 2x2 buildings, the quadrambles of service walkers generated by 3x3 buildings (Table 3) were inspected for the shortest lags at which discontinuities could be identified. The right side of Fig. 2 shows 3x3 buildings at their "go-blind" positions. The asymmetry of vision of 3x3 buildings is even more pronounced than was the case for 2x2 buildings. A 3x3 building in the north quadrant of a four-way intersection does not seem to be able to clearly see the intersection even if the intersection is only three squares away, but a 3x3 building in the south quadrant can see the intersection just as far away (six squares) as a 1x1 building can. East- and west-quadrant 3x3 buildings could only see the intersection up to four squares away. The implications that the north-south asymmetries in the go-blind distances of 2x2 and 3x3 hold for the shape of the region of the map examined most intensively by the quadramble-generating algorithm are explored in the discussion.

Figure 2. Buildings at the closest distances to four-way intersections at which they first "lose site" of the intersection and fail to dispatch their walkers on bracketing walks in clockwise order along all four roads. The grass shows the visible part of the footprint of a second 3x3 building in the north quadrant that overlapped with Seth's temple.

(Continues.)

Instability of quadrambles. In recording the quadrambles of roaming walkers from 2x2 buildings near four-way intersections, three examples of instability of walk direction were observed only for buildings located sufficiently far from the intersection so that their quadrambles had assumed either modified or pure straight-road form. In all three cases, one of the legs of the straight road quadramble that took the walker through the intersection could be modified to take the walker in either of just two directions. These directionally unstable walks are identified in Table 2 as two numbers in a quadramble separated by a vertical "OR" bar that I stole from programming languages like C that use the vertical bar for that operator. In addition, a priest from a temple in the south quadrant of an X intersection at an offset of -6i also contained one quadramble leg in which I saw directional instability (Table 3). After rechecking the quadrambles to identify the nearest lag from the intersection at which each size of building along each road at which modified straight road form was achieved I can now say that I never saw directional instability until a sufficient lag was reached for the quadramble to assume modified or pure straight road form.
   To finally see the directional instability of the 3x3 buildings quadramble indicated in Table, I had to march that priest around for several years, but it finally showed up. I do not now recall why I was so suspicious of the S-6i 3x3 building. In hindsite, if I had displayed similar patience when recording the quadrambles of 2x2 or 3x3 buildings at the first straight-road quadramble lag, I suspect I could have documented several additional examples of directional instability. I would never trust the directions indicated by the quadrambles in the tables presented here to remain rock-solid stable for any leg that took a walker through the intersection, if that walker came from a building far enough from the intersection that it could not "see" the intersection when generating its walker's quadramble. I would certainly never trust the stability of an entry in a quadramble directing a walker through a four-way intersection from a building that was far enough away from the intersection for its quadramble to have assumed pure or modified straight-road form.

Generalizability. Simultaneously to lessen any doubts about the reliability of the quadrambles that were extrapolated from the data gathered in Cleopatra's Alexandria (denoted by superscript a's in the tables) and to assess the extent to which the quadrambles from buildings that can "see" a four-way intersection can be relied upon to apply to other maps, I set up two lab towns: one in Rostja and the other in Men-Nefer (Yes, the very early mission that teaches you how to make papyrus, how to trade via land, and how to build a little mastaba. It offered a nice, big area of desert and no military action.) The town in Rostja never worked very well. I had to shoot water carriers in from the ends of the arms of my road X, which I could not do for the housing on the NE road without violating my no-bends-within-26-squares rule. Nevertheless, I got my sprites to plug some ostriches, and I managed to feed and water the populace enough to put everyone into ordinary cottages. Along the way, I managed to verify that the walkers from service buildings around the intersection positioned at several offsets for which the quadrambles had only been interpolated from Cleo data did, in fact, walk the interpolated quadrambles indicated in the tables by the superscripted R's. Not a single departure from the predicted quadrambles was observed in Rostja, but I had not had the heart to tax anybody yet.
   Men-nefer gave a much more satisfactory city. It also confirmed the interpolated quadrambles marked in the tables with superscript upper case M's. Even subject to the severe restrictions on road geometry that were required for experimental purposes, this city farmed grain and barley, dug clay, cut reeds in both reed beds, produced beer, pots, and papyrus (which it traded to Nekhen), put all citizens into common residences and built the little mastaba. Moreover, not one industry ever flickered off for want of labor (although I shut some down periodically because they were too ferociously productive). Nor did any service-providing industry lose its waving flag for lack of workers. I may only know about 200 quadrambles, but they certainly give a city designer an incredible sense of liberation. It is rather reassuring to know when to put the work camp above the road instead of below, because the quadrambles are wrong on the other side and it will have poor access to labor.
   All the interpolated quadrambles within sight of the four-way intersection proved rock solid. Consquently, I could cluster all the service providers (except entertainers) within known-quadramble space about that intersection and meet the needs of my populace. Most short-walking (i.e., 34-square bracketing, 26-square defaulting) service providers could satisfy everyone on the X within their wanderings. The nice, steady clockwise rotation ensured that everyone got serviced in turn. Some homes even in this mostly desert landscape got malaria before an apothecary was added near the intersection, but never after. No housing ever devolved for want of a physician. However, the teacher was clearly just shy of the maximum extent of the amount of road that he could service before devolution occurred.
   I only had two overlapping positions at which I could locate my water supply in Men-Nefer. At first, my water carrier was clearly not up to the task. Housing would devolve at about the time that he reached the 10th square of the road leading up to the thirsty people, who all resided at least 14 squares from the intersection. Since he was so close, I tried boosting the water supply with a large statue (which induced road pavement), all the plaza that would fit within its range or effect from the water supply, one small statue and one garden. That fixed things; no more houses went thirsty - ever. Unfortunately, all that desirability also goosed the recently placed bazaar into evolved status, so I cannot say whether one green girl alone can adequately meet the needs for food and dry goods of an X community. Two certainly can.
   One miserable bandstand on the main four-way intersection of an X community does not even come close to meeting the public's needs for entertainment, if they are to be housed in common residences. A pavilion might suffice. According to VitruviusAIA's housing chart or its predecessor by Grumpus the Elder, common residences need 30 entertainment points to stay stable. A small housing block could obtain this from a bandstand with frequent passage by a juggler (10 pts.) AND frequent passage by a musician (20 pts.). In an X village, this simply is not going to happen often enough; the juggler from the bandstand is off on his clockwise cycle and so is the musician (somewhere else), but with only a single bandstand drawing entertainers from one juggler's school and a conservatory, there was never close to enough entertainment. I built bandstands at the ends of the three roads that did not have a conservatory on them, and that fixed the problem. Of course, a single dancer gives 30 entertainment points just by walking by, so if only common residences are needed, a pavilion on the intersection might do the job. I could not check. Men-Nefer does not allow dancers.
   Long walkers are the big problem for an X village. One fireman unambiguously fails to meet its needs. Two firemen can, almost, if their walks are very nearly out of phase (180 out of phase) from one another ("I turn around in the Northeast at exactly the same moment you reverse course on the southwest road"). Firemen can easily be placed out of phase by saving the game and exploiting initial recruiter invariance, as described in a reply to my earlier posting (StephAmon 26DEC01). Two architects placed out of phase kept all my storage yards, bricklayers guilds, temples, clay pits, and one granary standing without a single collapse. Two out-of-phase fire wardens did not afford similarly flawless protection to combustible structures, particularly to those located far out on the roads from the intersection. However, three fireguys timed to be 120° apart did the job, nicely.
   The magistrate also presented a problem, if he was allowed to travel along all four roads as far as he wanted. Making those huge 51 square bracketing walks simply takes too long to allow him to visit all the housing often enough to prevent devolution back to common apartments. Once all the quadrambles had been checked for the long walkers, I road blocked off the ends of the roads to shorten his travels and that stabilized things nicely. The other vexation with the magistrate was that he seemed to need to pass a bit more housing than most other walkers to prevent his building from spitting out a labor recruiter after every one to three walks. This recruiter emission caused the magistrate to skip the leg of his quadramble that the recruiter had just used and, thus, to neglect one road's housing, which promptly devolved to apartments. Three common residences on each of the four roads were insufficient to suppress labor recruiter emission by this courthouse, but four common residences per road were enough.
   The tax collector gave me a nasty turn when I started recording his turn-around squares. He was most definitely not walking to the predicted squares in the quadrambles plus an extension factor of 17 squares, like the other long walkers were. But he was not turning around where a short walker from a building substituted for the tax office would either. Once I had the complete quadramble, I could see that the tax guy was adding an extension factor like the true long walkers, but that his was only nine squares instead of 17. So, a tax collector appears to be a kind of "medium walker".

(Continues.)

[This message has been edited by StephAmon (edited 01-17-2002 @ 01:54 AM).]

Discussion

I believe we have enough information to deduce the existence of an area of known size and shape around most walker-generating buildings (other than the really big ones like Palaces and temple complexes) within which the road geometry has special importance in determining many properties of the quadrambles of the walkers from those buildings.  To conserve space, I propose the term "domain of strong road influence" or just "domain" for this area.  Although even the data presented here contain evidence of the quadramble-generating algorithm's ability to "see" roads and intersections outside a building's domain, it often appears not to bother looking.

To deduce the size and shape of a building's domain, requires that we know the kind of information the algorithm gets when it "looks" at a road square within its domain.  The designers probably had to set up at least one huge (oddly shaped) two-dimensional array of some kind to hold the map and all the roads and structures we add.  Each position in this map array probably holds a standard data structure that identifies what is present in the corresponding position on the map.  The data structure undoubtedly has to include at least one position for a pointer to accommodate buildings like storage yards for which a great deal of data needs to be stored to fully describe it and all its contents, special instructions it has been given, perhaps information about its staffing status, and where its cart pushers are: potentially a lot of stuff.  What we need to know is: What kind of information does a routine "see instantly" when it takes a look at a road square?  The collection of data needed to describe a road square has got to be a lot simpler than for a storage yard.  Is it dirt, paved, or plazaed?  What we really care about is whether or not the structure of data about a road square specifies whether it has connections to the four neighboring squares.  I propose that we make the best guess we can about the answer to this question, and see whether we can determine the size and shape of building domains based on that guess.  If we guess incorrectly, we are likely to find out that we cannot deduce domain shapes using our first guess.

My best guess was that the information stored for a road square that a routine within the program can see just by looking at the data stored for the map array entry for its map position does include information on which neighboring squares the examined road square connects to by road.  It would only take four bits of storage space in the road square's data structure to specify yes (1) or no (0) about whether road connections exist to the NE, SE, SW, and NW.  Including those four bits of information would make life a whole lot easier for anyone who had to code routines that looked at road squares.  If you want to redraw the screen, for example, you could just go through the big map array one square at a time and fill in the screen with the road squares showing the appropriate shape.  We all know that the map shows road squares differently if they are part of straight roads, T-junctions, isolated road squares, and the like.  If the connection information is stored in the master array, a routine looking at a road square does not need to also look at all surrounding square's just to find out those four bits of information before it can display the road square.  This would also make life easier for the poor soul who had to write the FindShortestRoadPath() between-two-points routine that the game undoubtedly requires.  This assumption matters because it means that if the road square representing the four-way intersection falls just barely within a building's domain, the quadramble generating routine will "see" the four-way intersection even if it cannot "see" all four roads within its domain.

Hypotheses winnowed by tests of blindness.  We can get a good sense of the margins of the domain's of buildings of different sizes, by looking at the first offsets from the intersection along all four road at which different sizes of buildings first show discontinuities in the patterns within which they had been changing their quadrambles as a function of increasing offset distance.  For the 1x1 buildings, this kind of "loss of clear vision" occurred in every direction at a lag of six when the intersection was seven squares away.  Fig. 3 shows two possible "candidate" shapes for a building's domain that would see a four-way intersection six squares away but not seven. These are proposed as starting hypotheses which we will try to falsify and/or modify in repeating cycles until we arrive at a shape that survives our efforts to disprove it.  The square (desert) model on the left side of Fig. 3 survived the first test: it sees not only the intersection but three of the roads as well. It cannot see the road to the northwest, but it does not need to, if the data structure stored in the map array for the intersection includes the information "I am connected by road to my northwest".  If the firehouse at the center of the square domain were moved just one square to the SW, the intersection would no longer lie within its domain and it would be "blind" to it.  If the firehouse with the square domain were positioned six squares from the intersection (at an offset with an absolute value of 5) along either side of any of the four roads it would see the intersection, but would lose sight of it if the building were moved just one square further away from the intersection.  So far, this hypothetical domain shape cannot be rejected.

Figure 3. Two starting candidates for the size and shape of the domain of strong road influence that contain the intersection six-squares away from 1x1 buildings. A "square" domain is shown on the right, and the architect's post at left is shown at the center of a "diamond" domain.

(Continues.)

[This message has been edited by StephAmon (edited 01-16-2002 @ 11:30 PM).]

The diamond-shaped (grass) model in Fig. 3 also looks like it would work. If the architect's post were shifted one square to the right, it would lose site of the intersection. If the architect's post were placed along either side of any of the four roads six squares from the intersection, with the diamond-shaped domain it would see the intersection, but would lose site of it if the building were moved one square along the road further from the intersection. Alas, there is a problem with the diamond model. If the architect's post in Fig. 3 were shifted right by one square, it would still be able to "see" the road square on the road to the southwest that was just below the intersection. It would also be able to see the road square just to the SE of the intersection. If these two squares contain information about their road connections to neighboring squares, the quadramble-generating algorithm would "know" that at least a corner (leading to a road to the southwest) existed at the spot actually occupied by the four-way intersection. In Table 1, the quadramble recorded for a building in the south quadrant at lag +6 (which is where the architect's post would be if shifted right by one square) is [40, 33, -28, 33]. None of the entries send the walker down the road to the southwest, as would have been the case if the -28 were turned into a -28i. The actually observed quadramble seems bizarre, if the algorithm could see a road to the SW on which it could be sure that it could put the building's walker but could not see the road to the NW where the walker actually goes. Exploration of Table 1 reveals several cases in which the algorithm sends walkers down roads that it would only know about if the four "vertex" squares of the diamond-shaped domain were actually part of the domain. However, such exploration also uncovers plenty of instances in which the algorithm fails to send walkers down the roads it would certainly know about if the vertex squares were part of a 1x1 building's domain and instead sends the walkers in directions it could not see within the diamond domain. Therefore, I rejected the diamond model in its original form.

We can keep a diamond-like model in the contest if we modify the original version slightly by cutting off the four vertex squares to produce a truncated diamond-shaped domain like that shown with grass around the firehouse in Fig. 4. I actually went around the diamond model to see for each vertex square if I could see evidence in the quadrambles for 1x1 buildings (Table 1) that would force me to exclude that square from the building's domain. The left side of Fig. 4 shows one more vertex square failing the test. The quadramble for a west quadrant dentist at the offset of -6 shown in the figure was [26, 19i, -42, 19i], which would make little sense; the only road (other than the one to the NW that the building was sitting on) that the algorithm could "see" within the building's domain happened to be the only one of the four roads along which it never sent its walker. Ridiculous! Another vertex square got cut off.

Figure 4. Two variants of the diamond domain. Retention of a vertex square on the desert domain at left allows the dentist to see the road to the southwest even at an offset of -6 from the intersection. The fully truncated diamond domain (in grass) allows the firehouse to "see" the intersection at an offset of 5 (as shown), but would cause the firehouse to lose site of of the intersection and all roads save the one on which it sits at an offset of 6.

(Continues.)

The next step in the hunt for the shape of a building's domain was to include the quadrambles for the 2x2 buildings in the winnowing of our two surviving candidate domain models. There seemed to be two logical ways in which the domain of a 1x1 building could be modified to produce an analogous domain for a 2x2 building:

i, the 1x1 domain shape could be preserved but its size could be somehow expanded so that it still lay symmetrically around a 2x2 building - extending out six squares to the NE, SE, SW, and NW; or

ii, both the size and shape of the original 1x1 domain could be preserved for a 2x2 building, and the 1x1 domain could centered around one square of the four covered by a 2x2 building.

We can immediately reject alternative i, because it would allow 2x2 buildings to "see" just as far to the SE and SW as they can to the NE and NW. We know that the locations of the discontinuities in the patterns of numbers in the quadrambles for 2x2 buildings occur at different "go-blind"distances for buildings in the south quadrant of a four-way intersection (which see the intersection six squares away) than for buildings in the north quadrant (which can only see the intersection four squares away). Longer vision from the south quadrant than from the north seemed to imply that the domain was centered about the north square of 2x2 buildings.

I test drove both the square domain centered about the north square of 2x2 buildings and the truncated diamond domain for buildings located along both sides of all four roads leading from the intersection to see if one of the models did a better job than the other of predicting the offsets at which discontinuities occurred within the quadrambles for 2x2 buildings (Table 2). The square domain model clearly flunked this test. It incorrectly predicted that buildings in the east and west quadrant would go blind at offsets (with absolute values) of 6 from the intersection (The actual go blind offsets were 5, 5i, and -5), and it incorrectly predicted that 2x2 buildings in the north quadrant would go blind at offsets of 5i and -5, whereas they actually go blind at 4i and -4. I was insufficiently clever to find a surgical modification of the square model that would allow it to pass this test. Alas, my preferred hypothesis augured in and was burned beyond recognition.

Fig. 5 shows the truncated diamond model centered around the north squares of two 2x2 buildings passing this test with flying colors. The school on the right side of Fig. 5 is at offset 5 and the water supply on the left is at offset -4. Neither building could "see" the intersection in a domain with the shape shown in the figure. Both buildings can see the road to the northeast, and they know that at least a corner occupies the position of the four-way intersection. The truncated diamond model around 2x2 buildings passed the test along both sides of all four roads.

Figure 5. Two 2x2 buildings at offsets from which they cannot see the four-way intersection within their domains but can see the road to the northeast. The truncated diamond domains are centered around the north square of each building.

(Continues.)

[This message has been edited by StephAmon (edited 01-17-2002 @ 00:05 AM).]