When True Is False
Photographs courtesy of Ilusionworks, L.L.C.



Impossible = Possible The sculpture above, by artist Mathieu Hamaekers, appears to be a triangle made with three straight beams. Viewed from a different angle (right), however, we see that the beams are actually curved. In Figure 1, below, a proof devised by computer scientist David Huffman reveals why it is impossible to make a triangle with three straight beams. Notice that there are three visible surfaces. Each is flat, so it lies on a plane. The edges where the planes intersect are shown in red. Consider how many different ways three planes in space can intersect. First, picture a corner where two walls and the floor meet. These three planes intersect in three lines, and these lines meet in a single point, the corner of the room. If three planes enclose a long, triangular tube, the planes intersect in three parallel lines. If two of three planes are parallel, the planes intersect in two parallel lines. And three parallel planes don't intersect at all. But in Figure 1, the red lines where planes intersect do not meet at a single point, nor are they parallel, so the figure is impossible.

Examine the eight other figures below. Can you determine which are geometrically impossible and which are merely improbable? Assume that seemingly straight lines really are straight, and all surfaces are perfectly flat. Surfaces that appear to meet at an edge do in fact meet, and they don't lie on the same plane. Exactly three surfaces meet at every corner. Be wary of the right angles, however. Appearances can be deceiving.





Black = White According to the thesaurus, black means white. Not directly, but by way of a chain of synonyms. We consulted www.thesaurus.com and learned that another word for black is bleak. Under the heading for bleak we found bare. Continuing along those lines, bare = clean = white. Can you construct similar chains of synonyms linking the word pairs below? For two words to count as synonyms, one must appear on www. thesaurus.com under the heading for the other word, and not as part of a phrase. A word's meaning needn't be the same for the two words it links. The word list below contains the missing links.


1.   lead = ______ = ______ = gold

2.   hot = ______ = cold

3.   slow = ______ = ______ = fast

4.   matter = ______ = ______ = energy

5.   necessity = ______ = ______ = invention

6.   true = ______ = ______ = ______ = false

7.   small = ______ = ______ = ______ = large

8.   positive = ______ = ______ = ______ = negative

9.   solid = ______ = ______ = ______ = ______ = ______ = liquid


Word List
biting
capital
cold
disinterested
fancy
fat
fluent
formidable
ideal
impersonal
mean
 mighty
moderate
oily
perfect
power
smooth
steady
thick
top
unreal
want
weight



1 = 2 Do you cringe at the sound of the word algebra? Do you have nightmares about factoring equations? Do you think it was all a pack of lies? If so, the two proofs at right seem to bear you out. Both "prove" that 1 = 2. But both are flawed. Can you spot the errors? (Hint: In Proof 1, try substituting numbers for the variables.) Proof 1
Assume a = b
Then ab = a2 = b2
So ab + (a2 - 2ab) = a2 + (a2 - 2ab)
Simplifying, a2 - ab = 2a2 - 2ab
Factoring, 1(a2 - ab) = 2(a2 - ab)
Therefore 1 = 2

Proof 2
Anything equals itself, so
1 + 1 - 1 + 1 - 1 . . . =
1 + 1 - 1 + 1 - 1 + . . .
Grouping pairs of terms, we get
1 + (1 - 1) + (1 - 1) + . . . =
1 + 1 + (-1 + 1) + (-1 + 1) . . .
Simplifying, we get
1 + (0) + (0) + . . . = 2 + (0) + (0) . . .
Therefore 1 = 2



Solution

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