Cycles of regular polygons

polycycle.sage

# http://math.stackexchange.com/q/1798630/35416

def polycycle(n, collected):
    cf = CyclotomicField(n)
    zeta = cf.gen()
    corner = -2*(1 + zeta)/(1 + zeta + 1 + 1/zeta)
    center = [corner*zeta^k for k in range(n)]
    gra = []
    for m in range(3, 7):
        mn = (m - 2)*n
        if mn & 1: continue
        for part in Partitions(mn//2, length=m):
            for perm in set(map(tuple, CyclicPermutations(part))):
                z = 0
                for k in perm:
                    z = z + 1
                    z = -z * zeta^k
                if z: continue
                # print(perm)
                corners = []
                overlap = False
                for k in perm:
                    f = lambda p: -(p + 2) * zeta^k
                    corners = center + map(f, corners)
                    if any(CC(c*c.conjugate()).real() < 1 for c in corners):
                        overlap = True
                        break
                if overlap: continue
                polys = Graphics()
                for k, poly in enumerate(matrix(cf, m, n, corners).rows()):
                    polys += polygon2d(map(CC, poly),
                                       title="{} regular {}-gons".format(m, n),
                                       color=hue(float(k/float(m)), 0.5, 0.5),
                                       edgecolor="black")
                polys.axes(False)
                gra.append(polys)
        if gra:
            print "{} regular {}-gons".format(m, n)
            break
    collected.append(gra)

collected = []
for i in range(3, 25):
    polycycle(i, collected)
maxlen = max(map(len, collected))
for row in collected:
    while len(row) < maxlen:
        row.append(Graphics())
graphics_array(collected).show(axes=False, title=None)