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Teaching Algebraic Concepts usingMathematicaIvanCnopVrije Universiteit Brussel, Brussels, elgiumicnop@vnet3.vub.ac.be

AbstractS t a t e of the art computer algebra packages allow treatment ofmathematical concepts beyond those which are t r a d i t i o n a l l y offered orp r e d e f i n e d . This i s important f o r mathematics education since suchconcepts are p r e c i s e l y those which were scarcely understood by studentsboth in secondary and tertiar y education. Implementation of thesemathematical concepts w i l l g r e a t l y improve t h e p o ss i b i l i t i e s f o rmathematics education at these le ve ls. Examples of such concepts are:

handling of i n e q u a l i t i e s ; l i m its, continuity, Lipschitz conditions and smoothness; growth of functions and the speed with which l i m i t values are

approached; quotients f s t r u c t u r e s ; spatial geometry and v i s u a l i s a t i o n of objects in up to four

dimensions.Examples of t h i s methodology are given using Mathematica, Quotientsa r e handled i n d e t a i l a n d t hi s pa r t i n c l u d e s examples o f t h e Mathematicacode used. These can be made a c t i v e by pasting in Mathematicanotebook format.

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92 Innovation In Mathematics1 Limit behavior and estimates

Many concepts in a n a l y s i s (and in most other s u b f i e l d s ofmathematics, except perhaps finite mathematics) involve "change" andsome kind of l i m i t i n g process. This makes them difficult to understand formany students both in secondary and tertiar y education. owadays suchl i m i t i n g processes can be simulated using animations a v a i l a b l e incomputer algebra packages running on all state of the art personalcomputers. I n t h e implementation o f such simulations i t i s necessary t oavoid creating artifacts due to the fmiteness of both soft- and hardware.The user should be a b l e to choose the speed and other s ett i n g s involvedi n the simulation, and should be a b l e to create his own examplesaccording to the same template.The methodology was explained in [1C].

The c e n t r a l i d e a is that of an estimate. hat do "getting small","ar b itrar i l y small", "getting large", "gets eventually bigger than", ...mean? The animations or simulations ave to be set up in such a wayt h a t while viewing, the user gets an i d e a of the i n e q u a l i t i e s involved. Thise l i m i n a t e s the fmiteness of the computer setup s i n c e the viewer extracts,out of the finite sequence of frames, the p r o p e r t i e s of the objects t h a tremain v a l i d o v e r a l l . By looking at the p i c t u r e s , he gets h i n t s on makingconjectures and on how to prove the corresponding theorems. To someextent, this process could be automated. Translating estimates of severalkinds i n t o r u l es , theorems from mathematical a n a l y s i s can be obtained.L i t t l e work seems to have been done here.

2 Quotients

2.1 Predefined quotientsUnderstanding quotients poses severe problems for the average

student. This is the case even for d i s c r e t e and finite structures. A quotienti s a set of equivalence c l a s s e s under some equivalence r e l a t i o n . In somecases the quotient i n he r i ts the operations on the o r i g i n a l structure and thep r o p e r t i e s thereof. Here color coding can h e l p understanding howq u o t i e n t s work. Such color codings are q u i t e n a t u r a l and r e a d i l y a v a i l a b l ei n theMathematica documentation. The Color wheel is the most obviousexample: its Hues correspond to equal arguments and saturationcorresponds t o moduli. Coloring p o l a r coordinates i s t h e simplestexample of coloring in the ComplexMap Package. Along the same idea, aContourPlot or DensityPlot of a function shows the equivalence r e l a t i n gp o i n t s with nearby values by coloring or shades of grey. In this case l i t t l ea l g e b r a i c o r geometric s t r u c t u r e i s involved, except f o r some very s pe c i a lfunctions.

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Innovation In Mathematics 93

In other cases, i t i s p o s s i b l e t o g e t useful information b y coloringobjects in the r i g h t way. Coloring a sphere or a torus according tomeridians (longitude) and l a t i t u d e s explains how they r e l a te to rectanglesi n the plane. These colorings are done using the parameters in theParmetricPlot command. Spherical coordinates blow up near the poles ona sphere. They do not blow up on a torus. Thus we f i nd t h a t i d e n t i f y i n gopposite s i d e s of a rectangle with periodic coloring, this rectangle can ber o l l e d i n t o a tube which can then be turned i n t o a torus. Rolling irst neither d i r e c t i o n gives the same final result. If one tries to do a similarcoloring on the Klein Bottle according to the longitude and l a t i t u d eparameters i t w i l l lead t o t h e surprising r e s u l t that i n o n e direction th ec o l o r i n g must be symmetric around the axis. The portion next to th is axisi s turned i n t o a Mobius strip.

S u b s t a n t i a l information about finite groups can be derived froma p p r o p r i a t e colorings of the geometric objects under consideration.Choosing an appropriate coloring of the Dodecahedron exhibits ani n s c r i b e d cube (or an inscribed tetrahedron) and enables us to investigatewhich rotations of the Dodecahedron preserve the inscribed cube (or thei n s c r i b e d tetrahedron), showing i n t e r e s t i n g subgroups. Rotations or orderfive preserve n e i t h e r of these. Coloring the four diagonals of the cube w i l lthen show that mappings of a cube onto itself are a l l permutations on fourelements. This brings us to the a b s t r a c t s ett i n g of permutation groups.

2.2 Quotients in abstract inite roupsAbstract groups require more care. Here is a setup of input l i n e s for

o b t a i n i n g quotients by a subgroup in the group of permutations on a setof n elements:

obj ects=Table[j, {j,n}];perms=Permutations[objects];with i t s composition l a w

comp[perm2_,perml_]:=Table[perm2[[perml[[j]]]],{j,n}]S e l e c t i n g a nonempty set f elements in the permutation group

indices={ (some list of indices) }sub=Table[perms[[indices[[j]]]],{j,Length[indices]}]one has to v e r i f y t h a t these permutations form a subgroup by checkingthat t h e composition l a w i s i n t e r n a l (unit element a n d inverses come f o rfree in the case of i n i t e roups):subGroupQ[sub_]:=

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94 Innovation In MathematicsUnion[ Flatten[ Outer[comp,sub,sub,1],1] ]==sub

I f this r e t u r n s False, one can look for the subgroup generated by thesubset by n e s t i n g outer [ comp [ ] ] or f in d i n g i t s FixedPoint :

generl[sub_]:=Union[sub,Flatten[Outer[comp,sub,sub,1],1] ]generated[sub_]:=FixedPoint[generl,sub]a n d r e p l a c i n g s u b b y this generated [sub] I t i s n o w p o s s i b l e t o f i n da l l left cosets o f this subgroup

coset[j_] :=Flatten[Outer[comp,{perms[[ ]] ]} ,sub,1],1]which may be r e p e a t e d . Next one has to r e o r d e r the elements accordingt o t h e p a r t i t i o n i n t o c o s e t s , which a r e added o n e b y o n e only i f i t selements were not a l r e a d y p r e s e n t

add[j_]:=Sort[Flatten[Table[Position[perms,coset[j][[k]]],{k,Length[sub]}]]]after the o r i g i n a l subgroup. We o b t a i n an ordering of the elements in thegroup which reflects t h e p a r t i t i o n i n t o c o s e t s :

l=indices;order=Do[1=1f[Union[Join[1,add[j]]]==Union[l],1,Join[1,add[j]] ],{j,Factorial[n]}];1This issomewhat t r i c k y s i n c e checking e q u a l i t y of cosets r e q u i r e s s o r ti n gelements, a n d t h e elements i n t he final l i s t should never b e sorted b y t h el e x i c o g r a p hi c o r d e r i n g b u i l t i n t o Mathematica.Now the c o l o r i n g of elements is s t r a i g h t f o r w a r d :

colors[m_]:=Hue [N[Floor[(m-1)/Length[sub]]/Length[perms]Length[sub]]]I f t h e order o f elements i s taken t o b e t h a t o f t h e l i s t 1 :

mult [ p o s l _ , p o s 2 _ _ ] : =First[First[Position[perms, comp[perms[[l[[posl]] ]],perms[[1[[ pos2]] ]] ] ]]m[posl_,pos2_]:=First[First[Position[1,mult[posl,pos2]]]]

F i n a l l y a l l elements ar e c o l o r e d :g=Table[colors[m[posl,pos2]],{posl,Length[1]},{pos2,Length[1]};Show[Graphics[RasterArray[g]]]

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Innovation In Mathematics 95It may be s u r p r i s i n g that we did not check the normality of the subgroup.I f t h e subgroup i s n o t normal this RasterArray i s messy s i n c em u l t i p l i c a t i o n o f t h e cosets i s n o t independent o f their representingelements. The subgroup is normal i f and only i f the RasterArray shows aneat square block structure. In many cases i t w i l l be easy to recognise thestructure of the quotient roup by this p a t t e r n .

In case of the 24 permutations on n=4 objects, with the Kleinfour-group its subgroup, allelements are ordered and colored i n t o 6colors, (which for the sake of p r i n t i n g of this te xt were replaced bycorresponding GrayLevel [ ] s). M u l t i p l i c a t i o n is done according to thefollowing table.

The elements of the Klein four-group are shown in the b r i g h t e s t tone.T h e m u l t i p l i c a t i o n t a b l e o f t h e Klein four-group i s i n t he upper leftcorner. It is easy to recognise this 6 by 6 block t a b l e as the m u l t i p l i c a t i o nt a b l e of permutations on 3 elements. The whole t a b l e is real l y a 24 by 24square ta b l e. This fact can be made c l e a r by adding some options whilerendering the Graphics: first d e f i n e a t i c k s g r i d and then see what comeso u t i f o n e adds t h e following options i n show:

grid=Table[i,{i,0,Length[perms]}];Show[Graphics[RasterArray[g]],GridLines->{grid,grid},AspectRatio->Automatic]I f we have a subgroup of allpermutations or a group defined Ianother w a y i t i s sufficient t o r e p l a c e i n t he above i n p u t l i n e s "perms" b y

t h e l i s t of elements in the group and "comp" by the law of compositionunder consideration. For i n s t a n c e we can take a d i h e d r a l group, i.e. thegroup of mappings of a r e g u l a r n-gon, and i t s composition law:

di2Group[n_]:=Flatten[Table[{j,ref1},{j,0,n-1),{refl,0,!}],!]comp[{j_,ref11_J ,{k_,ref!2_}] : =

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96 Innovation In MathematicsIf[refll==0,{Mod[j+k,n],ref!2},{Mod[j-k,n],Mod[refll+refl2,2]}]

In case we have a l l movements of a regular ctagon, we s pe c i f yn = 8;perms = di2Group[n];

together with some subset sub= {{4,0}, {4,1}} and continue inputst o generate a (hopefully normal) subgroup and obtain the quotient.2.3 Recognition of the quotients

The above reasoning can handle r a t h e r large groups although it maynot be easy to see what the s t r u c t u r e of a quotient is in the case of alargegroup with a small subgroup. The number of blocks w i l l be large and thecoloring w i l l be somewhat hazy. Different techniques w i l l be necessary torecognize the structure of the q u o t i e n t . One p o s s i b i l i t y is to take asubgroup of the q u o t i e n t and try to get some information by repeating theq u o t i e n t operation, thus mimicking the theory of solvable groups. Acascade of q u o t i e n t s may give the Jordan decomposition.

Another technique is to consider the order of the elements. The orderof an element in a group is d e f i n e d by the number of elements in thesubgroupgenerated b y i t:

order[element_]:= Length[generated[{element} ]a n d h as t o b e extended t o a s i m i l a r d e f i n i t i o n f o r cosets using t h ed ef i n i t i o n of cosets above. If the orders of the cosets do not match witht h e orders of the elements in a known group, these groups are notisomorphic. Thus the only groups we have to compare with are thosewith the same physionomy of orders of elements. It is easy to b u i l d asorted l i s t o f t h e orders o f all elements.

I f necessary we s t i l l have to permute the elements in the knowngroup, but only permuting among the elements which have the sameo r d e r s . This should n o t b e t o o difficult, a n d t h e number o f p o s s i b i l i t i e s i snot too big in many cases.

It i s a l so p o s si b l e t o make u p a m u l t i p l i c a t i o n t a b l e with a l l roupelements named (with colored text) in a s i m i l a r way according to thesame p a r ti t i o n i n t o cosets. This however w i l l put r e s t r i c t i o n s on the sizeof the groups under c o n s i d e r a t i o n and a lot f formatting has to be donet o avoid misaligned ta b l es . Misaligned t a b l e s a r e l i ke l y t o occur i fdifferent string lengths a r e involved i n t he naming o f group elements.

F i n a l l y , the set of i n p u t s could be organised i n t o a Package form ifs uff i c i e n t i n t e r es t develops. I t c a n f u r the r b e extended i n t o a n e l e c t r o n i chigher algebra course, but I would p r e f e r such a project to be r e a l i s e d byspe c i a l i s t s i n t he field. They have a t th e i r d i s p o s a l a l o t o f theorems which

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Innovation In Mathematics 97

can be implemented. They w i l l find h i n t s for programming in the abovei n p u t lines.

I would refrain from s ta r t i n g such a program for several reasons.First, special-purpose packages f o r doing abstract algebra i n groupsalready exist a n d i t m a y b e difficult t o achieve their q u a l i t y withoutsustained input of manpower over a longer period of time. Secondly,c r e a t i n g too many special purpose packages may end up hiding the mainadvantage o f Mathematica , i.e. i t s universal u s a b i l i t y . Last b u t n o t least:most of the fun resides in doing the exercise of getting mathematical ideasorganised in the r i g h t way so that Mathematica can handle them.

This is also what we finally hope to achieve with our students: thatthey understand t h e mathematics suffic ie nt l y well i n order t o develop th issetup and make discoveries for themselves, rather than s u b s t i t u t i n gv a r i a b l e s for blank spaces in some ready-made function or package.

3 Geometric objects in 3 and 4 dimensionsIt is known that applying the show command with different

viewpoints enables the viewer to fly around three dimensional objects,thus efficiently v i s u a l i s i n g them, even f o r t h e inexperienced. This i simplemented by the SpinShow command, although more elaborateways o f f l y i n g around a n object remain p o s s i b l e . What i s less c l e a r i s thatfour dimensional objects can be ef f e c t i v e l y "seen" using an animation.The graphics g a l l e r y in the Mathematica book [SW] offers severalexamples o f surfaces i n four dimensions fo r which i t i s difficult t ounderstand th e i r true nature using only a three dimensional perspective i np r i n t e d form. T h e Klein Bottle i n t he graphics g a l l e r y does n o t self-i n t e r se c t a n d i s therefore b e s t viewed b y a n animated sequenceh i g h l i g h ti n g a p a r t o f it t h a t keeps being shifted. T h e same i s true f o r a l lexamples of Riemann surfaces in the book.

One should further i n v e s t i g a t e the p o s s i b i l i t y of using stereoscopicviews f o r constructions i n three a n d four dimensions.

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98 Innovation In Mathematics4 Conclusion

The unifying i d e a behind such implementations of seemingly d i ffe r e n tt o p i c s i s t o offer t o t he mathematics l e a r n e r s a n environment which i sh i g h l y v i s u a l , but which does not hide the mathematical content of what ispresented. Having at our d i s p o s a l such techniques w i l l enhance i n t e r es t in mathematics as a t o p i c improve retention, and speed up the learning curve facilitate the task of the teacher.It seems es s e n t i a l t o real ise th is i n a short term given t h e increasinglydifficult s i t u a t i o n o f mathematics i n education i n many countries.

References

[1C] I. Cnop: Teaching a b s t r a c t concepts by animation, Proceedings oft h e F i r s t Asian Technology Conference in Mathematics,Singapore, 1995.

[SW] S. Wolfram: TheMathematica Book, 3 " *e d. , Wolfram Media &Cambridge Univ Press, 1996.