The Most Irrational Number

Here is the TrueBasic program that generated the images of seed packing.
! seed packing

LET s = 400
! set window size - large to make room for type
SET WINDOW 0,s,0,s
LET g = .5*(5^(.5)+1) ! set spacing factor
!LET g = sqr(2)
!LET g = pi
!LET g = 7/31
LET n = 80 ! set number of seeds
LET a = 30 ! set size of (eqil triang) seed
LET i = 1 ! initialize
LET m = a/2
DIM h(200) ! NOTE this must be at least n

DO while i < n+1
LET x = (i*g - int(i*g)) * s !calculate ith x-coordinate
LET gim = x - m
IF gim < 0 THEN ! fix to be mod s
LET gim = gim + s
END IF
LET gip = x + m
IF gip > s THEN ! fix to be mod s
LET gip = gip - s
END IF
LET h(i) = m*sqr(3) !h(i) will be height of APEX
LET j = 1
DO while j < i
LET xj = (j*g - int(j*g)) * s
LET gjm = xj - m
IF gjm < 0 THEN
LET gjm = gjm + s ! fix to be mod s
END IF
LET gjp = xj + m
IF gjp > s THEN ! fix to be mod s
LET gjp = gjp - s
END IF
CALL hij( gim, gip, gjm, gjp)
IF hnew > h(i) THEN
LET h(i) = hnew
END IF
LET j = j + 1
LOOP
CALL tri ( x, h(i))
LET i = i + 1
Pause .1
LOOP

SUB hij(gim, gip, gjm, gjp) !correction to height i to miss triangle j
CALL dif (gip, gjm) !determine overlap between base of
IF d < 0 THEN !i-th triangle and base of j-th
LET hnew = m*sqr(3)
EXIT SUB
END IF
CALL dif (gip, gjm + m)
IF d < 0 THEN
LET hnew = h(j) + sqr(3)*(gip-gjm)
EXIT SUB
END IF
CALL dif (gip, gjp)
IF d < 0 THEN
LET hnew = h(j) + m*sqr(3)
EXIT SUB
END IF
CALL DIF (gim, gjm + m)
IF d < 0 THEN
LET hnew = h(j) + m*sqr(3)
EXIT SUB
END IF
CALL DIF (gim,gjp)
IF d < 0 THEN
LET hnew = h(j) - sqr(3)*(gim-gjp)
EXIT SUB
END IF
LET hnew = m*sqr(3)
END SUB

SUB dif (a, b) !calculate distances mod s
LET d = a - b
IF d < -s/2 THEN
LET d = d + s
EXIT SUB
END IF
IF d > s/2 THEN
LET d = d - s
EXIT SUB
END IF
END SUB

SUB tri (a,b) !draw triangle and plot i inside
PLOT a-m,b-m*sqr(3);a,b
PLOT a,b;a+m,b-m*sqr(3)
PLOT a-m,b-m*sqr(3); a+m,b-m*sqr(3)
PLOT TEXT, at a-8,b-m*sqr(3)+2: str\$(i)
END SUB
GET key zz
END

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