# A cuboid that can be bisected into two cuboids, all three being similar (sides ratio $1:2^{\frac{1}{3}}:2^{\frac{2}{3}}$): any book references, etc.?

In the Euclidean plane, one can define the so-called root-2 rectangle, i.e. a rectangle whose side lengths are in the ratio $1 : \sqrt{2}$, or 1 : 1.414 (3 d.p.); a key property is of being divisible, with a straight-line cut, into two rectangles which are similar to the first, i.e. side lengths in the ratio $1 : \sqrt{2}$ (this property is the basis of a commonly used standardized paper format, e.g. A4 paper). The "A"series of paper sizes.

Going into 3-D Euclidean space, I've found an object with the same property: a cuboid with side lengths in the ratio $1: 2^{\frac{1}{3}} : 2^{\frac{2}{3}}$. When bisected with a planar cut as shown in the figure, the two identical cuboids created are similar to the original (one set of similar faces are shown in blue).

Having googled, I've so far found no mention of this elegantly simple cuboid that features the Delian constant, i.e. $2^{\frac{1}{3}}$. Does anyone know of references to this object (book, web-based, etc.)? ## 1 Answer

This is a 2-rep brick, a 3D rep-tile. I looked at this back on March 2 with W. R. Somsky, and meant to ask about it here. It was new to me then. With a root of $x^4-x^3-1=0$, about x=1.38028, three similar bricks can make a non-similar cuboid. But that was as far as I got. • – Many thanks for the answer and rep-tile link. Looking at your first figure, I see now, of course, that the cuboids produced from the first cut (in my figure) can themselves be bisected to produce two cuboids similar to the original, and so on...(which graphics package, by the way?). These objects also nicely illustrate how geometrical constructions yield polynomial equations (such as the quartic equation obtained from your second geometrical construction); going in the other direction, I wonder how many polynomials have yet to be identified with elegant geometrical constructions? – Jimbo Jun 20 '18 at 6:44
• – Ed Pegg Jun 20 '18 at 12:58
• Delian Brick sounds like a good name for it. – Ed Pegg Jun 20 '18 at 14:22
• The "Delian Brick" it shall be - I hope the name catches on! – Jimbo Jun 20 '18 at 20:34