COEKESPONDENCB
333
To the Editor, The Mathematical Gazette
DBAB SIB.—The point made by Mr. Dunn at the end of his article
" Tessellations with Pentagons " (Gazette LV, NO. 394 (December 1971),
pp. 366-9) about the reversal of the basic condition for a pentagon to be
a tessellating cell by having "2 of the angles adding up to 360° and the
other 3 to 180°" is interesting. Outlined below is a method of constructing such a pentagon, which will necessarily be re-entrant.
FIG.
1
Draw any re-entrant par-hexagon ABCDEF (that is, a hexagon with
three pairs of parallel sides), and produce any pair of parallel sides,
EF and BO say, by equal distances inward to points P, Q respectively, as
in Figure 1. Join PQ.
Because a par-hexagon enjoys point symmetry about its centre O, we
have here that PO — OQ. We have thus divided the hexagon into 2
congruent pentagons (the one can be obtained from the other by a halfturn about 0), which can now be used to tessellate a plane. I t will be
noticed that because EP and BQ are parallel,
LFPQ = LPQC;
therefore, in the pentagon
ABQPFA,
reflex L FPQ + L PQB = 360°,
and it is easily seen that the other three angles of the pentagon sum to
180°. (It is well-known that any par-hexagon can be used to tessellate
a plane surface.)
All that the above construction really does is to divide the parallel
sides EF and BG externally in the same ratio. I t is interesting to note
that if we divide the parallel sides of any par-hexagon internally in the
same ratio, as in Figure 2, we have the first type of pentagons, with 2
adjacent angles supplementary, here angles FPQ and BQP.
The
pentagons so formed are (i) re-entrant or (ii) convex according as the
par-hexagon is re-entrant or convex.
Downloaded from https:/www.cambridge.org/core. IP address: 88.99.165.207, on 15 Jun 2017 at 18:38:25, subject to the
Cambridge Core terms of use, available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/S002555720013116X
334
THE MATHEMATICAL GAZETTE
FIG.
2
I t should also b e n o t e d t h a t a s P a n d Q t r a v e l r o u n d t h e sides of t h e
p a r - h e x a g o n , p r o v i d e d a l w a y s t h a t t h e line s e g m e n t PQ is wholly
c o n t a i n e d w i t h i n t h e h e x a g o n , a whole class of p e n t a g o n a l cells is
g e n e r a t e d , each of w h i c h c a n be u s e d for tessellation.
Cameroon College of Arts, Science and Technology,
Bambili, Mezam
Division,
N.W.
Cameroon
Y o u r s faithfully,
P. NSANDA E B A
[Mr. E b a also d r a w s a t t e n t i o n t o Mr. D u n n ' s second class of tessell a t i n g p e n t a g o n s , of w h i c h a s p e c i m e n is r e p r o d u c e d below. I n t h e
original article it is s t a t e d t h a t " each h e x a g o n is m a d e u p of four
p e n t a g o n s , t w o of which a r e m i r r o r images of t h e o t h e r t w o " . Mr.
D u n n p o i n t s o u t t h a t h e here uses t h e description " m i r r o r images " t o
describe p e n t a g o n s r e l a t e d b y a n opposite i s o m e t r y , b u t n o t of course b y
a single reflection. H i s p o i n t is t h a t t h i s tessellation includes c o n g r u e n t
non-regular a s y m m e t r i c a l p e n t a g o n s (I a n d I I I in t h e figure) a n d t h e
s a m e p e n t a g o n s " t u r n e d over " ( I I a n d I V ) , n o t j u s t r o t a t e d in t h e
p l a n e . H e r e m a r k s " I t h i n k t h i s is fairly u n u s u a l " . Mr E b a a d d s t h a t
I a n d I I I a r e o b t a i n e d from each o t h e r b y h a l f - t u r n a b o u t t h e centre of
t h e h e x a g o n ( m a r k e d w i t h a cross in t h e figure), as are I I a n d I V . T h e
c o n s t r u c t i o n of t h i s tessellation can b e r e c o m m e n d e d as a n i n s t r u c t i v e
a n d e n t e r t a i n i n g exercise.
The fundamental pentagon: a = a', b = b ' and x + x ' = 180°.
Downloaded from https:/www.cambridge.org/core. IP address: 88.99.165.207, on 15 Jun 2017 at 18:38:25, subject to the
Cambridge Core terms of use, available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/S002555720013116X
CORRESPONDENCE
335
Tessellation of pentagon forming a pattern of interlocking hexagons.
Finally, we are grateful to Mr. Eba for pointing out that the statement
in the last line of p. 368, that the pentiamond has " all the sides equal "
is, of course, incorrect; actually, three of the sides are 1 unit in length
and the other two are 2 units each. D.A.Q.]
GLEANINGS FAR AND NEAR
The following comments on relative motion were made at the trial of
Reginald Tom Hinks for the murder of his father-in-law on 1st December,
1933, and are quoted by F. Tennyson Jesse in "Comments on Cain", 1948.
When Dr. Scott-White, a witness for the defence, said he thought the
bruise more consistent with a moving object striking a stationary object
than a stationary head being met by a moving object, the learned Judge
merely asked "Why?" Dr. Scott-White replied: "May I put it this way?
Would you rather I hit you on the head with an ink-pot or would you
rather fall on the ink-pot?" To which the learned Judge replied: "So
long as the strength of the blow is the same I don't think it would matter."
(per Mr. A. B. Manning)
Downloaded from https:/www.cambridge.org/core. IP address: 88.99.165.207, on 15 Jun 2017 at 18:38:25, subject to the
Cambridge Core terms of use, available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/S002555720013116X