In the 1841 edition of "Elementary Geometry with Applications in
Mensuration," Charles Davies refers to parallelopipedons (pages 142-147) in
Book VI, Of Solids.
-- "Theorem X: Two regular parallelopipedons, having equal altitudes and
equal bases, are equal."
-- "Theorem XI. Two regular parallelopipedons, which have the same base, are
to each other as to their altitudes."
-- "Theorem XII. Two regular parallelopipedons, having the same altitude, are
to each other as to their bases."
-- "Theorem XIII. Any two rectangular parallelopipedons are to each other as
the products of their three dimensions."
-- "Theorem XIV. If a parallelopipedon, a prism and a cylinder, have
equivalent bases and equal altitudes, they will be equivalent."
The "Concise Oxford Dictionary of Mathematics" defines a "parallelepiped (a
word commonly misspelled)" as "a polyhedron with six faces, each of which is
a parallelogram." Webster's Ninth New Collegiate Dictionary (not always
accurate in these matters) places the word's origin in the year 1570.
Of the dozen or so other basic geometry texts of more recent origin that I
have on hand, not one mentions the word, parallelepiped. (let alone the more
aged usage, parallelepipedon, and its variant spellings).
Searches on Hotbot and Yahoo turned up fewer than a dozen Web references to
the parallelepiped with the majority of them employing the word to describe
the shape of an object such as the "Bvlgari Quadrato" wristwatch, but more
often by architects in the Mediterranean, Chile, and Japan describing the
shape of buildings.
Example 1, <http://www.ime.gr/fhw/en/projects/3d/xanthos/building_g.html>,
has the following reference: "The building "G" is located at the west side of
the acropolis, adjoined to the south side of the surrounding wall and is
visible from the town. It is the biggest from the three buildings of worship
with monumental dimensions. It consisted of a rectangular, artificial terrace
(ground plan 15,5x10,25 metres and height 2,35 metres) and upon it rose the
main building of worship (ground plan 6,60x4,25 metres and height 5 metres).
It was all made of stone, had the shape of a rectangular parallelepipedon
with a flat roof and a projecting horizontal cornice."
See also <http://www.autobank.it/Puglia/crg/virtual/08/wrt12.htm> (this and
the following URL didn't connect when I tried them a few days later): "the
Ognissanti Church . . . what remains of a little Benedictine monastic complex
founded by Eustasius." Another site similarly describes a late 15th century
church design by Leonardo da Vinci.
The only mathematical reference I found was an abstract at
<www.kilin.u-shizuoka-ken.9c.ip/prof/form96.html>: Two Topics on Plane
Tiling, Hiroshi Fukuda, Toshiaki Betumiya, Shizuka Nishiyama, and Gisaku
Nakamura:
"The paper consists of two topics on plane tiling, i.e., tilings with
glide-reflection (pg tiling) of polyominoes and a solid model for Penrose
tiling, that were separately presented at the conference. The former
investigates some mathematical characters on pg tiling and obtains all
polyominoes, from domino (2-omino) to 11-omino, having the pg tiling by
exhaustive computer search. It is interesting to note that some polyominoes
have several different types of pg tiling. The latter investigates the solid
model obtained by recursive substitutions of star rhombic dodecahedra with
the size of rp-3, where rp(=1.618034) is the golden ratio, and finds that
Penrose tiling is obtained by a single parallelepipedon, the acute
rhombohedra usually denoted by A6. The result may require some modifications
of the theory previously established, in which two kinds of parallelepipeda,
the acute and obtuse rhombohedra usually denoted by A6 and O6, must be
provided."
Finally, digging into DejaNews I found "Dr. Math" expounding on the subject:
<<From: "John Conway" <conway@math.Princeton.EDU>
Subject: Re: 11-gon
Date: Mon, 24 Oct 94 20:10:39 EDT
<snip>
I mentioned "ped" because it happens in that curious word
"parallelepiped", which should really be pronounced parallel-epi-ped. Until
about the middle of the 19th century this word was even longer -
"parallelepipedon". It splits into parts thus
para - allele - epi - ped - on
beside other upon ground
(memo - "pedon", meaning the ground, is what you put your foot on) Two things
are "parallel" if one is beside the other - this was already used as a single
word very early on. What the name means is that there's always a face that's
parallel to the one upon the ground.>>
All of this brings to mind a few questions in the context of this quick look
at a fading word.
1. Parallelepipedon seems to have fallen almost completely from usage in the
United States sometime between 1850 and 1930. Does anyone have a more
precise knowledge of its last "common" usage here? Or where it may be extant?
2. Are there instances in which the word or its variants are actually found
in modern (ok, post-modern for some folks) geometry textbooks at the primary,
secondary or early university level in the United States? Elsewhere?
3. Where parallelepiped usage is extant in parts of Europe, South 'America
and Japan, does the concept survive in geometry courses, or is its usage
limited to the jargon of architecture or arcane mathematical discussions?
David Coia
Arlington, VA