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The name Fakataka can be read as faka-taka, 'forming' (haka) a 'cycle' (taka):

Taka

Taka, takataka. Circle; to form circles, to gather, to get together (of people). Vanaga.

1. A dredge. P Mgv.: akataka, to fish all day or all night with the line, to throw the fishing line here and there. This can only apply to some sort of net used in fishing. We find in Samoa ta'ā a small fishing line, Tonga taka the short line attached to fish hooks, Futuna taka-taka a fishing party of women in the reef pools (net), Maori takā the thread by which the fishhook is fastened to the line, Hawaii kaa in the same sense, Marquesas takako a badly spun thread, Mangareva takara a thread for fastening the bait on the hook. 2. Ruddy. 3. Wheel, arch; takataka, ball, spherical, round, circle, oval, to roll in a circle, wheel, circular piece of wood, around; miro takataka, bush; haga takataka, to disjoin; hakatakataka, to round, to concentrate. P Pau.: fakatakataka, to whirl around. Mq.: taka, to gird. Ta.: taa, circular piece which connects the frame of a house. Churchill.

Takai, a curl, to tie; takaikai, to lace up; takaitakai, to coil. P Pau.: takai, a ball, to tie. Mgv.: takai, a circle, ring, hoop, to go around a thing. Mq.: takai, to voyage around. Ta.: taai, to make into a ball, to attach. Churchill.

In other words, Woman unites the generations of Men. The Old Sun (Kui) drowns and his newborn baby boy will be the son of a widow.

Kui

Ku'iku'i 1. To disturb, to inconvenience, to feel uncomfortable, said for instance of a thief who has hidden the things stolen under his clothes: he-ku'iku'i i roto i a îa te me'e toke, the stolen things inconvenience him; he-ku'iku'i te vânaga-haga, his manner of talking betrays embarrassment. 2. To crowd together; he-ku'iku'i te gagata i te uruga mai ki te hare, the people are crowding to get into the house. Vanaga.

To see T. Kuikui, to stagger. Churchill.

Tui

1. To sew mats, to make strings. E-tahi tuitui reipá i Te Pei, ekó rava'a e-varu kaukau; i-garo ai i Hiva, i te kaiga, a necklace of mother-of-pearl is on te Pei, few will find it (lit: eight groups of people); it has remained in Hiva, in our homeland. 2. The three stars of Orion's Belt. Vanaga.

Allen had his Barani, the 'bearer' (of a new generation), as lunar station number 28, but the end and the beginning are close together (like α and ω), and by the force of precession it will take only about 2 * 13 * 71 = ca 1846 years to move ahead 2 places in the list of stations.

28 * 13 = 364 = 26 * 14.

1842 A.D. - 26 * 71 = 5 B.C.

April 15 16 17 (107) 365
October 15 16 17 (290)
Ca1-25 Ca1-26 Ca2-1 (27)
kiore ki te huaga kua moe ki te tai. Te heke
no star listed POLARIS, Baten Kaitos (26.6), Metallah (26.9) Al Sharatain-1 / Ashvini-1 / Bond-16
 Segin, Mesarthim, ψ Phoenicis (27.2), SHERATAN, φ Phoenicis (27.4)
67 A.D. 5 B.C. 76 B.C.
τ Bootis (208.3), Benetnash (208.5) ν Centauri (208.7), μ Centauri, υ Bootis (208.8) Muphrid (210.1), ζ Centauri (210.3

In Popol Vuh is hinted at how Woman - described as a phase of the Moon - is like a vessel (a 'canoe') across from one generation to the next:

... And then the bone spoke; it was there in the fork of the tree: Why do you want a mere bone, a round thing in the branches of a tree? said the head of One Hunaphu when it spoke to the maiden. You don't want it, she was told. I do want it, said the maiden. Very well. Stretch out your right hand here, so I can see it, said the bone.

Yes, said the maiden. She stretched out her right hand, up there in front of the bone. And then the bone spit out its saliva, which landed squarely in the hand of the maiden. And then she looked in her hand, she inspected it right away, but the bone's saliva wasn't in her hand. It is just a sign I have given you, my saliva, my spittle. This, my head, has nothing on it - just bone, nothing of meat. It's just the same with the head of a great lord: it's just the flesh that makes his face look good. And when he dies, people get frightened by his bones. After that, his son is like his saliva, his spittle, in his being, whether it be the son of a lord or the son of a craftsman, an orator.

The father does not disappear, but goes on being fulfilled. Neither dimmed nor destroyed is the face of a lord, a warrior, craftsman, an orator. Rather, he will leave his daughters and sons. So it is that I have done likewise through you. Now go up there on the face of the earth; you will not die. Keep the word. So be it, said the head of One and Seven Hunaphu - they were of one mind when they did it ...