November 2020 - mathfun@mailman.xmission.com

[math-fun] Rolling polyhedra
by James Propp 26 Dec '20

26 Dec '20

I want to think about rolling polyhedra, and the polyhedra I want to think about rolling are polyhedra in which the sum of the angles at each vertex is a submultiple of 360 degrees. For instance, given any triangle T, we can create a tetrahedron whose four faces are all congruent to T, with angle-sum 180 degrees at each vertex (there's a name for such tetrahedra but I forget what it is). Are there other polyhedra in which the angles at each vertex sum to a submultiple of 360 degrees (not necessarily the same one at each vertex)? Jim Propp

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[math-fun] Klein bottle puzzles
by Dan Asimov 29 Nov '20

29 Nov '20

Let K denote a Klein bottle. 1. The cartesian product S^1 x S^2 of a circle (S^1) and a sphere (S^2) is a certain 3-dimensional manifold. Puzzle: Does this manifold contain a Klein bottle K ⊂ S^1 x S^2 as a subset ??? 2. Call two simple closed curves C_0, C_1 on the Klein bottle "equivalent" if there is a continuous family {C_t ⊂ K | 0 ≤ t ≤ 1} of simple closed curves on K. How many inequivalent simple closed curves are there on K, and what is an example of each one ??? —Dan Appendix -------- A surface is called "nonorientable" if it has a subset that is a Möbius band. Otherwise it is called "orientable". The (compact) orientable and nonorientable surfaces (without boundary) were first classified up to topological equivalence by August Ferdinand Möbius in 1861, but this was not proved rigorously until Henry Roy Brahana did so in 1921. Any two surfaces can be combined topologically by the operation called "connected sum", which means to remove the interior of a disk from each surface and then connect the resulting two circular boundaries by a cylinder. Each orientable surface M is the connected sum of g tori (where the connected sum of 0 tori is defined to be the sphere). Each nonorientable surface M is the connected sum of g projective planes for some integer g ≥ 1, also called the "genus" of M. If S denotes the sphere, T denotes the torus and P denotes the projective plane, then every surface (compact and without boundary) can be expressed as the connected sum of S with finitely many copies of T and finitely many copies of P. This operation, denoted by #, has S as the identity element and is commutative with just one relation: P # P # P = P + T. It turns out that each compact surface M can be "triangulated" — expressed as the union of finitely many 2D triangles any two of which intersect in a common edge or vertex. Then if V, E, F denote the number of vertices, edges, and faces (triangles) of the surface, the computation 𝜒(M) = V - E + F is a topological invariant called the Euler characteristic of M. If M is orientable its genus satisfies g = 2 - 2 𝜒(M), and if M is nonorientable its genus satisfies g = 2 - 𝜒(M). Thus Euler characteristic and orientability determine the surface up to topological equivalence. The Klein bottle K shows up in this classification as K = P # P, the connected sum of two projective planes. It can also be viewed as the result of starting with a square and identifying the top and bottom sides in the same direction, but identifying the left and right sides in the reverse direction: |—————➞————| | | ↑ // ↓ | | |—————➞————|

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[math-fun] "MALAYALAM" isn't just the longest palindrome in my length 400000 wordlist;
by Bill Gosper 27 Nov '20

27 Nov '20

It's the longest word whose reverse is a word: In[282]:= MaximalBy[DeleteCases[words, "MALAYALAM"], If[StringQ@# && MemberQ[words, Evaluate[StringReverse@#]], StringLength@#, 0] &] /@Range@6// tim During evaluation of In[282]:= 4166.771102,6 Out[282]= {"DESSERTS", "DIORAMAS", "REDRAWER", "REWARDER", "SAMAROID", "STRESSED"} W/o the Evaluate this took over 2.5 hrs. In[295]:= WordDefinition@"samaroid" Out[295]= {"pain caused by maple seeds at or inside the anal sphincter. Relieved by Preparation S."} —Bill

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[math-fun] NestList. For when you get bored and keep pushing the same buttons on your calculator.
by Bill Gosper 27 Nov '20

27 Nov '20

In[290]:= Integrate[Cos@x/(x^2+1),{x,-∞,∞}] Out[290]= π/*e* In[291]:= ContinuedFraction[%,105] Out[291]= {1,6,2,2,1,2,6,8,2,1,1,1,4,3,1,1,66,2,1,1,2,4,2,7,46,10,2,1,18,1,23,10,14,1,1,4,3,1,6,2,5,1,1,1,18,355, 1,1,5,3,4,1,28,5,2,1,3,16,3,25,4,7,1,11,2,2,1,9,2,31,1,20,1,2,1,1,6,13,1,3,1,3,4,125,1,7,2,2,1,3,5,1,1,6,1,1,1,9,1,1,4,6,4,1,13} (Prove it's endless; Win a Fields Medal.) In[292]:= {GeometricMean@%} (*In[295]:= N@Khinchin Out[295]= 2.68545200106531*) Out[292]= {2^(8/15) 3^(23/105) 5^(13/105) 7^(1/21) 2201^(1/105) 3289^(2/105)} In[293]:= NestList[(Print[N@#];GeometricMean@ContinuedFraction [#,105])&,%[[-1]],9] During evaluation of In[293]:= 3.09506434681568 During evaluation of In[293]:= 2.41520362449092 During evaluation of In[293]:= 2.9335577403418 During evaluation of In[293]:= 2.44930689361668 During evaluation of In[293]:= 2.71218585275826 During evaluation of In[293]:= 2.6450717647845 During evaluation of In[293]:= 2.77752000086026 During evaluation of In[293]:= 2.74022410644692 During evaluation of In[293]:= 2.59352990789751 Out[293]= {2^(8/15) 3^(23/105) 5^(13/105) 7^(1/21) 2201^(1/105) 3289^(2/105),2^(18/35) 3^(29/105) 5^(1/35) 104977873^(1/105),2^(19/35) 3^(22/105) 5^(3/35) 7^(2/21) 11^(2/105) 39497^(1/105),2^(7/15) 3^(4/21) 5^(8/105) 7^(2/35) 11^(1/35) 589^(1/105),2^(7/15) 3^(2/7) 5^(8/105) 7^(2/35) 19^(2/105) 1639^(1/105),2^(64/105) 3^(5/21) 5^(8/105) 209^(2/105) 851^(1/105),2^(53/105) 3^(4/21) 5^(11/105) 13^(4/105) 77^(1/35) 1919^(1/105),2^(47/105) 3^(6/35) 5^(11/105) 11^(1/35) 91^(2/105) 316238983^(1/105),2^(3/5) 3^(8/35) 5^(2/35) 7^(4/105) 11^(1/35) 221^(1/105),2^(43/105) 3^(13/105) 35^(8/105) 242280181^(1/105)} In[294]:= NestList[(Print[N@#];GeometricMean@ContinuedFraction [#,105])&,%[[-1]],42] During evaluation of In[294]:= 2.39797710382867 During evaluation of In[294]:= 2.69893228647732 During evaluation of In[294]:= 3.29061256432359 During evaluation of In[294]:= 2.03162608678672 During evaluation of In[294]:= 3.01906752481969 During evaluation of In[294]:= 2.59538767809429 During evaluation of In[294]:= 2.68778297494002 During evaluation of In[294]:= 3.11804540780474 During evaluation of In[294]:= 2.98027331397385 During evaluation of In[294]:= 2.23842419582078 During evaluation of In[294]:= 2.73195233677031 During evaluation of In[294]:= 2.32112558219677 During evaluation of In[294]:= 2.65266653473585 During evaluation of In[294]:= 2.45995266400843 During evaluation of In[294]:= 1.94020851734991 During evaluation of In[294]:= 2.13798184316968 During evaluation of In[294]:= 2.75100265543326 During evaluation of In[294]:= 3.56350709479461 During evaluation of In[294]:= 2.59571502601519 During evaluation of In[294]:= 2.4740594814946 During evaluation of In[294]:= 2.38638422095896 During evaluation of In[294]:= 2.94605839370828 During evaluation of In[294]:= 2.82204016953752 During evaluation of In[294]:= 2.25028325915228 During evaluation of In[294]:= 2.36248747987146 During evaluation of In[294]:= 2.39582513918321 During evaluation of In[294]:= 2.79821884430297 During evaluation of In[294]:= 2.51021506839255 During evaluation of In[294]:= 2.52467515174716 During evaluation of In[294]:= 2.40158175513891 During evaluation of In[294]:= 2.54575104382523 During evaluation of In[294]:= 2.70499590456546 During evaluation of In[294]:= 2.36456538650387 During evaluation of In[294]:= 2.82338870219823 During evaluation of In[294]:= 2.33636891461969 During evaluation of In[294]:= 2.4610503767373 During evaluation of In[294]:= 2.50128322134356 During evaluation of In[294]:= 2.43209873810393 During evaluation of In[294]:= 2.83712666091056 During evaluation of In[294]:= 2.92217685976039 During evaluation of In[294]:= 2.58135001107047 During evaluation of In[294]:= 2.97981675199117 Out[294]= {2^(43/105) 3^(13/105) 35^(8/105) 242280181^(1/105),2^(7/15) 3^(16/105) 5^(1/21) 77^(1/35) 54503796054221^(1/105),2^(58/105) 3^(41/105) 5^(1/15) 17^(2/105) 77^(4/105) 247^(1/105),2^(43/105) 3^(17/105) 5^(2/35) 133^(2/105) 671^(1/105),2^(11/21) 3^(4/21) 5^(8/105) 7^(2/35) 17^(1/35) 437^(1/105) 4433^(2/105),2^(58/105) 3^(29/105) 13^(4/105) 17^(1/35) 11165^(1/105),2^(7/15) 3^(19/105) 5^(3/35) 19^(1/35) 2639^(2/105) 19987^(1/105),2^(3/5) 3^(8/35) 17^(1/35) 35^(1/21) 209^(2/105) 245401^(1/105),2^(58/105) 3^(4/15) 11^(1/35) 35^(4/105) 1417^(2/105) 2369^(1/105),2^(41/105) 3^(16/105) 5^(1/15) 7^(2/105) 11^(1/35) 11497861^(1/105),2^(46/105) 3^(4/15) 5^(13/105) 7^(4/105) 1431859^(1/105),2^(17/35) 3^(1/5) 5^(1/21) 91^(2/105) 410533^(1/105),2^(58/105) 3^(19/105) 11^(1/35) 35^(2/35) 374153^(1/105),2^(59/105) 3^(23/105) 35^(1/21) 39121^(1/105),2^(18/35) 3^(2/21) 5^(2/35) 17^(1/105) 77^(2/105),2^(13/35) 3^(4/21) 5^(8/105) 11^(1/35) 44863^(1/105),2^(19/35) 3^(8/35) 5^(2/35) 7^(4/105) 11^(2/105) 13^(1/35) 34523^(1/105),2^(22/35) 3^(8/35) 5^(1/15) 7^(2/35) 17^(1/35) 143^(2/105) 460112447^(1/105),2^(31/105) 3^(32/105) 5^(13/105) 7^(1/21) 11^(1/35) 17^(2/105),2^(11/21) 3^(19/105) 5^(1/15) 19^(2/105) 77^(1/35) 377^(1/105),2^(41/105) 3^(9/35) 5^(4/105) 17^(1/35) 77^(2/105) 15067^(1/105),2^(22/35) 3^(26/105) 5^(1/15) 7^(4/105) 11^(2/35) 299^(1/105),2^(18/35) 3^(23/105) 5^(13/105) 17^(1/35) 77^(2/105) 3379^(1/105),2^(3/7) 3^(23/105) 5^(1/21) 7^(1/15) 1133^(1/105),2^(16/35) 3^(6/35) 11^(1/35) 13^(2/105) 35^(1/21) 1241^(1/105),2^(3/7) 3^(4/21) 5^(1/21) 11^(1/35) 13^(2/105) 80741659^(1/105),2^(3/7) 3^(16/105) 31^(2/105) 35^(1/15) 221^(1/35) 82777^(1/105),2^(8/21) 3^(9/35) 5^(1/21) 11^(1/35) 26739058073^(1/105),2^(16/35) 3^(5/21) 7^(1/15) 11^(1/35) 115^(2/105) 493^(1/105),2^(18/35) 3^(23/105) 5^(1/21) 7^(1/35) 19^(2/105) 13651^(1/105),2^(7/15) 3^(23/105) 11^(1/35) 35^(1/15) 899^(1/105),2^(62/105) 3^(19/105) 5^(4/105) 7^(1/21) 247^(2/105) 694331^(1/105),2^(41/105) 3^(2/15) 5^(2/21) 7^(1/21) 13^(4/105) 35563^(1/105),2^(18/35) 3^(6/35) 5^(8/105) 7^(2/105) 377^(1/35) 29932163^(1/105),2^(37/105) 3^(19/105) 5^(1/15) 7^(2/35) 11^(2/105) 2800733^(1/105),2^(41/105) 3^(1/7) 5^(3/35) 7^(1/21) 112819455463^(1/105),2^(10/21) 3^(6/35) 5^(1/15) 7^(1/35) 221^(2/105) 1123441^(1/105),2^(8/15) 3^(1/7) 5^(1/15) 7^(1/21) 24845483^(1/105),2^(53/105) 3^(26/105) 5^(8/105) 7^(2/35) 338981071^(1/105),2^(47/105) 3^(29/105) 5^(4/35) 7^(3/35) 13^(2/105) 493^(1/105),2^(46/105) 3^(19/105) 5^(8/105) 77^(1/35) 1202287073^(1/105),2^(64/105) 3^(5/21) 7^(4/105) 55^(1/15) 1079^(1/105),2^(61/105) 3^(6/35) 5^(1/15) 7^(1/21) 13^(2/105) 3023603^(1/105)} Methinks this could have a largish period. Limit? LUB? GLB? Tighter for larger values of "105"? —rwg

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[math-fun] denesting paradox?
by Bill Gosper 26 Nov '20

26 Nov '20

Are we still looking for the fabled case where neither √A nor √B denests, but √A+√B does? Unfortunately, this isn't a clear cut case, because the denesters go off the rails instead of declaring impossibility. Even the numerics is shaky. Out[19]= Sqrt[5] + Sqrt[7] == Sqrt[4 + (-54 + 35 I Sqrt[3])^(1/3)/3^(2/3) + 13/(3 (-54 + 35 I Sqrt[3]))^(1/3)] + \[Sqrt]((1/(162 I + 105 Sqrt[3]))(1296 I + 840 Sqrt[3] - 35 3^(5/6) (-54 + 35 I Sqrt[3])^(1/3) - 54 I (3 (-54 + 35 I Sqrt[3]))^(1/3) + 13 I (3 (-54 + 35 I Sqrt[3]))^(2/3))) Try to denest this RHS with Swastik's new package: In[20]:= ResourceFunction["RadicalDenest"] /@ %[[2]] During evaluation of In[20]:= Solve::ifun: Inverse functions are being used by Solve, so some solutions may not be found; use Reduce for complete solution information. During evaluation of In[20]:= Solve::ifun: Inverse functions are being used by Solve, so some solutions may not be found; use Reduce for complete solution information. Out[20]= Sqrt[ 4 + (-54 + 35 I Sqrt[3])^(1/3)/3^(2/3) + 13/(3 (-54 + 35 I Sqrt[3]))^( 1/3)] + \[Sqrt]((1/( 162 I + 105 Sqrt[3]))(1296 I + 840 Sqrt[3] - 35 3^(5/6) (-54 + 35 I Sqrt[3])^(1/3) - 54 I (3 (-54 + 35 I Sqrt[3]))^(1/3) + 13 I (3 (-54 + 35 I Sqrt[3]))^(2/3))) And with Corey's original package: In[49]:= $MaxExtraPrecision = 999; Strad /@ %20 During evaluation of In[49]:= N::meprec: Internal precision limit $MaxExtraPrecision = 999.` reached while evaluating -(105/13) Cos[2/3 (\[Pi]-ArcTan[Times[<<2>>]])]+3 Sqrt[13] Sin[1/3 (\[Pi]-ArcTan[Times[<<2>>]])]-54/13 Sqrt[3] Sin[2/3 (\[Pi]-ArcTan[Times[<<2>>]])]. During evaluation of In[49]:= MinimalPolynomial::nalg: (1/(Sqrt[6] 13^(1/4)))Cos[1/6 (\[Pi]+2 ArcTan[Times[<<2>>]])]^(\[Pi]/2+ArcTan[35/(18 Sqrt[3])]) Sqrt[(26 Sqrt[13]+<<1>>+(67+35 I Power[<<2>>]) Sin[1/3 <<6>>[<<1>>]]) <<1>>] is not an explicit algebraic number. What's all this trigamarole? In[50]:= %20 == \[Sqrt]5 + \[Sqrt]7 Out[50]= Sqrt[ 4 + (-54 + 35 I Sqrt[3])^(1/3)/3^(2/3) + 13/(3 (-54 + 35 I Sqrt[3]))^( 1/3)] + \[Sqrt]((1/( 162 I + 105 Sqrt[3]))(1296 I + 840 Sqrt[3] - 35 3^(5/6) (-54 + 35 I Sqrt[3])^(1/3) - 54 I (3 (-54 + 35 I Sqrt[3]))^(1/3) + 13 I (3 (-54 + 35 I Sqrt[3]))^(2/3))) == Sqrt[5] + Sqrt[7] In[51]:= N@% Out[51]= False Well, no wonder it didn't denest! Wait a minute, In[52]:= N[%%, 22] Out[52]= True That was a false negative! —rwg

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[math-fun] Hits when concatenating?
by Éric Angelini 24 Nov '20

24 Nov '20

Hello Math-Fun Say we add the last odd digit of S to a(n): S=1,2,3,6,9,18,19,28,37,44,51,52,57,64,71,72, 79,88,97,104,105,110,111,112,113,116,117,124,... Say we have a hit when the concatenation of two or more initial terms of S reappear in S; will we ever find a hit here (the hit « 123 » was missed by a single unit with « 124 », in the above example; the next possibilities would be with 1236, 12369, 1236918,... but the margin of this post, etc. Best, É.

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Re: [math-fun] "MALAYALAM" isn't just the longest palindrome
by Don Reble 24 Nov '20

24 Nov '20

> ... > 15 dermatoglyphics > 14 hydromagnetics > 14 hydropneumatic * > ... > * Plans for a gun with hydropneumatic brakes figured in the > Dreyfus Affair. So... the other words are uncopyrightables? -- Don Reble djr(a)nk.ca

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Re: [math-fun] "MALAYALAM" isn't just the longest palindrome
by Steve Witham 23 Nov '20

23 Nov '20

> From: Dan Asimov <dasimov(a)earthlink.net> > Date: 11/22/20, 1:21 AM > > Speaking of word oddities, Ararat (or tarara, as in "tarara boom-de-ay") has one T, two R's, and three A's. > > Is there any other word having its letter counts form an arithmetic sequence of at least three positive terms? > > Â—Dan In my dictionary, there are 204 words with counts (1, 2, 3), and 14 with (1, 2, 3, 4): 4 (1, 2, 3, 4) beerbibber Other than a potion in "Kingdom Come: Deliverance" and a multifunction bottle opener, see ** below. 4 (1, 2, 3, 4) chachalaca American tropical bird. 4 (1, 2, 3, 4) deadheaded 4 (1, 2, 3, 4) isoosmosis 4 (1, 2, 3, 4) kinnikinic Native American tobacco substitute. 4 (1, 2, 3, 4) kotukutuku The world's largest fuchsia. 4 (1, 2, 3, 4) peppertree 4 (1, 2, 3, 4) rememberer 4 (1, 2, 3, 4) sereneness 4 (1, 2, 3, 4) sleeveless 4 (1, 2, 3, 4) evennesses My dictionary tends to overdo suffixes. 4 (1, 2, 3, 4) keennesses 4 (1, 2, 3, 4) sanenesses 4 (1, 2, 3, 4) susurruses With two of each letter you get the arithmetic sequence 2, 2, 2, 2... (Computers can be so literal.) 5 (2, 2, 2, 2, 2) arraigning 5 (2, 2, 2, 2, 2) chimachima Yellow-headed caracara (bird) 6 (2, 2, 2, 2, 2, 2) happenchance 5 (2, 2, 2, 2, 2) horseshoer 5 (2, 2, 2, 2, 2) intestines 5 (2, 2, 2, 2, 2) nonordered 5 (2, 2, 2, 2, 2) quinaquina a.k.a. Cinchona plant yielding quinine. 7 (2, 2, 2, 2, 2, 2, 2) scintillescent scint i ll e scent 5 (2, 2, 2, 2, 2) strumstrum crude stringed instrument 5 (2, 2, 2, 2, 2) tromometer extra-sensitive seismometer Each of these words requires more than half the alphabet to spell: 14 (1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1) ambidextrously 14 (1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1) benzhydroxamic 14 (1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1) dermatoglyphic 15 (1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1) dermatoglyphics 14 (1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1) hydromagnetics 14 (1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1) hydropneumatic * 14 (1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1) pseudomythical 14 (1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1) subformatively 14 (1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1) sulphogermanic 14 (1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1) undiscoverably Nothing with at least three distinct letters and linearly increasing counts and more than one of the least common, nor more than four distinct letters. --Steve P.S. * Plans for a gun with hydropneumatic brakes figured in the Dreyfus Affair. ** beerbibber: The beginning of a 1927 Time Magazine article "Beer Diplomacy": To Riga, Latvia, came, some three months ago, Herr Doktor Adolph Koester, German Ambassador, lusty beerbibber. . . . He found official Latvia strongly disposed to favor the Franco-British project of a Baltic state cordon dividing Russia and Germany. Poland, close ally of France, had bestowed numerous decorations on Latvian officials. At Riga British diplomats were arbitri elegantiarum How should the German Ambassador win Latvian sympathies? Soon Herr Doktor Koester gave a true, orthodox German bier abend: a "beer evening" at which he and a group of high Latvian officials sat up with mossy... Many google hits like this: Spriest Driveadream beerbibber spriest.driveadream.com Â› beerbibber Yeah tie your fabric print. Telephony is dead. Rebel convoy is just taught his people where size does this door upon its hilltop. Somebody over here. Bases not ...

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Re: [math-fun] "MALAYALAM" isn't just the longest palindrome
by Keith F. Lynch 22 Nov '20

22 Nov '20

Tom Karzes <karzes(a)sonic.net> wrote: > I found two 10-letter (4 disctinct) words having this property: > sereneness > sleeveless Also, checking all words in /usr/share/dict/words on Panix, Actaeaceae, beerbibber, chachalaca, isoosmosis, kotukutuku, and rememberer. Next I'll check my saved email for more.

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Re: [math-fun] "MALAYALAM" isn't just the longest palindrome
by Stuart Anderson 22 Nov '20

22 Nov '20

> > That makes me wonder if there are long palindromic words in other > languages. > > If we allow words in a sentence to form palindromes, we get some long ones; Oozy rat in a sanitary zoo. Murder for a jar of red rum. Mr. Owl Ate My Metal Worm Napoleon Bonaparte: Able was I ere I saw Elba Are we not pure? “No, sir!” Panama’s moody Noriega brags. “It is garbage!” Irony dooms a man—a prisoner up to new era. The not so long but topical; Rise to vote, sir The maths problem; Never odd or even Then there's palindromic poetry; Doppelganger by James A. Lindon Entering the lonely house with my wife I saw him for the first time Peering furtively from behind a bush -- Blackness that moved, A shape amid the shadows, A momentary glimpse of gleaming eyes Revealed in the ragged moon. A closer look (he seemed to turn) might have Put him to flight forever -- I dared not (For reasons that I failed to understand), Though I knew I should act at once. I puzzled over it, hiding alone, Watching the woman as she neared the gate. He came, and I saw him crouching Night after night. Night after night He came, and I saw him crouching, Watching the woman as she neared the gate. I puzzled over it, hiding alone -- Though I knew I should act at once, For reasons that I failed to understand I dared not Put him to flight forever. A closer look (he seemed to turn) might have Revealed in the ragged moon. A momentary glimpse of gleaming eyes A shape amid the shadows, Blackness that moved. Peering furtively from behind a bush, I saw him for the first time, Entering the lonely house with my wife.

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