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SMILE!Logic smile!

An escalator can never break: it can only become stairs. You should never see an "Escalator Temporarily Out Of Order" sign, just "Escalator Temporarily Stairs. Sorry for the convenience"
      - Mitch Hedberg

•••

Always Look on the Bright Side of Life
(see the video!)

Some things in life are bad,
They can really make you mad,
Other things just make you swear and curse,
When you're chewing life's gristle,
Don't grumble,
Give a whistle
And this'll help things turn out for the best.
And...

Always look on the bright side of life.
[whistle]
Always look on the light side of life.
[whistle]

If life seems jolly rotten,
There's something you've forgotten,
And that's to laugh and smile and dance and sing.
When you're feeling in the dumps,
Don't be silly chumps.
Just purse your lips and whistle.
That's the thing.
And...

Always look on the bright side of life.
[whistle]
Always look on the right side of life,
[whistle]

For life is quite absurd
And death's the final word.
You must always face the curtain with a bow.
Forget about your sin.
Give the audience a grin.
Enjoy it. It's your last chance, anyhow.
So,...

Always look on the bright side of death,
[whistle]
Just before you draw your terminal breath.
[whistle]

Life's a piece of ****,
When you look at it.
Life's a laugh and death's a joke it's true.
You'll see it's all a show.
Keep 'em laughing as you go.
Just remember that the last laugh is on you.
And...

Always look on the bright side of life.
Always look on the right side of life.
[whistle]
Always look on the bright side of life!
[whistle]
Always look on the bright side of life!
[whistle]
Always look on the bright side of life!
[whistle]
Repeat to fade...

- Monty Pytons, 'Brian's Life'
(see the video!)

Always look on the bright side of life

 

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Previous Puzzles of the Month + Solutions

 
May-June 2008, Puzzle nr 117
arrow back Back to Puzzle-of-the-Month page | Home arrow home

 

Puzzle # 117 Difficulty level: bulbbulbbulbbulb, general math knowledge.

honeycomb honeycomb 2

The Geometry of the Bees...
  
The shape of the wax cells is such that two opposing honeycomb layers nest into each other, with each facet of the closed ends being shared by opposing cells, and with the open ends facing outward, as illustrated in fig. 1 above. Each cell is actually a rhombic decahedron, that is an hexagonal prism having three rhombi at its closed end (as shown in fig. 2 above). In short, each cell is a ten-sided structure with one side open. Mathematicians made extensive studies of the isoperimetric properties of the honeycomb cells and believed them to be the most efficient design possible. If the faces of every cell contain the LARGEST possible volume with the LEAST possible surface, what is the value of the angle alpha? The faces of the hexagonal prism are each 1-unit wide...

Key words: isoperimetric, rhombic decahedron, rhombus, hexagonal prism.

Related puzzles:
arrow Square vs rectangle problem.
arrow The best cardboard box.


Source of the puzzle:
©G. Sarcone, "Il Calcolo Differenziale ed Integrale", by Ing. Gustavo Bessière, Hoepli Publications, page 114.


solution
diagram solu 1 diagram solu 3
diagram solu 2

a) l is the hypotenuse of the right triangle MNP shown in fig. 3 above, then:
l2 = (2x)2 + 12 and
l = √(4x2 + 1)

b) The diagonal AB of the rhombus at the closed end of the cell does not depend upon its spatial angulation (the axis AB remains the same even when the spatial angle changes, fig. 4) but upon the sides of the hexagonal prism and the radius r that circumscribes the hexagonal section. Thus the length of AB is:
2√[12 - (1/2)2] = √(4 - 4/4) = √3

c) So, the area of each of the 3 rhombi at the closed end of the cell is:
dl/2 or
√3 √(4x2 + 1)/2

d) The area of each lateral trapezoid of the prism is:
1/2[h + (h - x)] · 1 unit, or
(2h - x)/2 units

e) Therefore, the whole surface of the cell is:
S = 3[√3 √(4x2 + 1)/2] + 6(2h - x)/2 units
Reducing to
S = 3/2[√3 √(4x2 + 1)] + 6h - 3x units

f) To know where the variable x peaks we take the derivative of S at the point 0:
S' = 3/2[8√3 / 2√(4x2 + 1)] - 3 = 0

so
6x(√3) / √(4x2 + 1) = 3
and finally
x2 = 1/8

g) If we replace x2 in the first equation a) with this last value, we obtain:
l = √(4x2 + 1) = √[4(1/8) + 1] = √1.5

The ratio of both diagonals of the rhombus gives us the tangent of the semi-angle α of the rhombus:
l/d = tg α/2 = √1.5/√3

Thus, the angle α of the rhombus at the closed end of the cell is:
2 arctg √1.5/√3 = 70.52877937.... ≈ 70° 31"
(arc = Inverse trigonometric functions)

 

cup winnerThe 5 Winners of the Puzzle of the Month are:
No winners... Yes, the puzzle was a little bit harder.

Sorry!


© 2004 G. Sarcone, www.archimedes-lab.org
You can re-use content from Archimedes’ Lab on the ONLY condition that you provide credit to the authors (© G. Sarcone and/or M.-J. Waeber) and a link back to our site. You CANNOT reproduce the content of this page for commercial purposes.

You're encouraged to expand and/or improve this article. Send your comments, feedback or suggestions to Gianni A. Sarcone. Thanks!


More Math Facts behind the puzzle

toth's structure  Curiously, a 3-dimensional honeycomb partition is not optimal! In 1965, the Hungarian mathematician László Fejes Tóth discovered that a cell end composed of two hexagons and two smaller rhombi (see fig. opposite), instead of three rhombi, would actually be 0.035% (or approximately 1 part per 2850) more efficient. But, frankly, this difference is too minute to measure on an actual honeycomb, and irrelevant to the hive economy in terms of efficient use of wax, and because the honeycomb walls have a definite thickness, it is not clear that Tóth's structure would indeed be an improvement... In that respect, the honeycomb is more like a wet foam than a dry foam. Several years ago, the physicist D. Weaire and his colleague R. Phelan undertook to construct two-layer foams with equal-sized bubbles, and they found that the dry foams did take on Tóth's pattern. But when they gradually added liquid to thicken the bubble walls, something 'quite dramatic' happened: the structure suddenly switched over to the three-rhombus configuration of a honeycomb (it seems, then, that the bees got it right after all!). The switch also occurs in the reverse direction as liquid is removed.



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