[These pages were authored by the late Thoki Yenn, and were restored here from the Internet Archive by Erik Demaine, with contributions of missing files from various readers (notably Tommy Stevens, Roberto Morassi, and Boaz Shuval). If you spot any other bugs like missing images or pages, please report them.]


More about The Diabolic Frame

Tommy Stevens has a miniature of this one on his Gallery 4

This three dimensional model is based on the special magic square known as the diabolic square. The properties of the diabolic square are remarkable enough, but the properties of this model are stranger still. All the numbers from 1 to 16 occur once only, but the groups of 4 add up to 34 in virtually every way you could imagine. Try adding up the four numbers on the inside, or on the outside, or on each face, or the numbers meeting at each corner on the inside, or each corner on the outside. Try starting anywhere and spiraling upwards or downwards. In every case the total is 34. If you search further, you can find yet more totals of 34. It seems astonishing that any diabolic frame can be made at all, but in fact 48 are possible.

7 12 1 14

2 13 8 11

16 3 10 5

9 6 15 4

Thok.91.( copied from der falter 13) Information taken from "MATHEMATICAL CURIOSITIES 1"

by Gerald Jenkins and Anne Wild. Tarquin Publications

Paulo Mulatinho is a good friend of Thok. He sent 75 copies of der falter 13 as a birthday present to Thok on his 75th birthday on 13, Jan, 1994

updated 15. August 2000

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The Diabolic Frame

Origami Version by Thoki Yenn

Original Design: Saint Pli

The numbers from 1-1 6 arranged on the principle of the Magic Square, on a square torus. Any group of four numbers add up to 34. Divide an A4 sheet in 5 sections along the middle line. Use the Fujimoto method or measure. Cut of 3/5 (12,6cm) and fold into 6 divisions like the diagram. Mark off the width of 3 sections along the long side, make a sharp crease, valley, and fold in the diagonals in the next section as indicated, mountains, leaving the small square in the middle without scars, but ringed by sharp mountain folds. Repeat the 3 width measure along the strip and make all creases very sharp. Repeat the treatment of diagonals and small square in section 4 from the right (this is also the section with the number 4 on it). Fill in the numbers. Fold like the profile, making a trial fold of two of the bends.Undo and form a flat ring by entering the short section without numbers into the section with the umber 1. The extra fold on that side will serve as a lock to avoid a loose flapping corner. Now wiggle the frame loosely back and forth in the shape of a parallelogram, while pressing lightly at the corners. If necessary, help the corners in forming by sticking a finger or a pencil inside. If you have given the corners a good precise trial bend, and sharp creases, it will click into a nice diabolic frame. The original version was folded from one piece of parchment with the proportions 1:2,7. 5 by 13,5 inches, (12,5 X 33,75cm). The original had only 5 sections, the sixth is an addition of mine in aid to the lock mentioned above. It was found crushed flat inside the bindings of an old illuminated manuscript on the island of Lindis Farne, East of North Cumberland, and is believed to have been made by the Irish monk Wolf de Eyre, who was later canonized as St. Pli.

If you are interested in "The Legend of St.Pli", write to Thoki Yenn.

Thok.93. Diagram and Description: Freising, May 1993.

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