top 10+ mathematical puzzles daniel … · cousin domes two different domes can be constructed from...
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TOP 10+ MATHEMATICAL
PUZZLES
DANIEL HARDISKY
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From:
Daniel J. Hardisky
2742 Superior Ave.
Parkville, MD 21234 USA
Retired Civil Engineer (1996, 2008) 69 years old and
Amateur Math Puzzle Guy.
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Cousin Domes
Two different domes can be constructed from an Octahedron (and/or
Cuboctahedron) breakdown. This is because two unique planes of symmetry exist,
producing two different hemispherical arrangements. One has an Octagonal base, the
other a Hexagonal base. Also, the Cuboctahedron has the amazing property that the
basic chord factor is 1.
A single kit can be used to construct either a Type-I dome with 28 pieces or a Type-
II dome with 27 pieces. If you get tired of one, re-assemble it into the other. The exact
same properties exist for both domes: chord factors, bevel angles and connectors. The
only difference is the arrangement of the members and connectors.
Type-I Dome: 12 Long, 16 Short, 5-4way & 8-6way connectors.
Type-II Dome: 15 Long, 12 Short, 3-4way & 9-6way connectors.
Chord Factors: Long pieces= 1.000 Short = 2*Sin(22.5º) = .76537
Bevel angles: Long = 30º Short = 22.5º
Connectors: 4way @90º (, from sph.∆’s)
6way: = arcos(1/√ 3 ) = 54º 44’
= arcos(1/3) = 70º 31’
The “kit” consists of: 15 Long, 16 short, 5-4way and 9-6way connectors. When
used as a tent, a single tarp could be sewn that incorporates slightly more than 1
hemisphere, and can be positioned differently for each type. These domes cannot be
built too large or the member lengths get too long and need to be stiffened. Probably
good for a radius of 4-6 ft. using ¾” conduit. Best used for children’s play-domes or tents.
The use of slightly more than 2 dozen pieces makes for easy assembly and storage.
The lines of symmetry can be seen on the magnetix models. Also the complete
sphere has: V+F=E+2 = 18+32= 50 = 48+2 as per Euler.
Daniel J. Hardisky
short
long
all short
long
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