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# Class Note for MATH 409 with Professor Martin at KU

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Notes on transformational geometry Judith Roitman Jeremy Martin April 22 2009 1 The intuition When we talk about transformations like re ection or rotation informally we think of moving an object in unmoving space For example in the following diagram when we say that the shaded triangle is the re ection of the unshaded triangle about the line Z we think about physically picking up the unshaded triangle and re ecting it about the liner This is not how mathematicians think of transformations To a mathematician it is space itself 2D or 3D or that is being transformed The shapes just go for the rider To understand how this works letls focus on the following basic transformations of the plane translations along a vector re ections about a line rotations by an angle about a point To help us consider these as transformations of the plane itself youlve been given two transparencies You ll keep one xed on your desk Youlll move the other one The one that moves represents what happens when you move the entire planet The one that stays xed tells you where you started fromi Project 1 Draw a dot on your paper Take a transparency sheet put it over your paper and trace the dot What can you do to the top transparency iiei plane so that the dots will still coincide lief which translations re ections and rotations leave the dot xed Now place two dots on the bottom sheet and trace them on the transparency The dots should be two different colors For the sake of discussion let s assume one is red and one is blue What can you do to the transparency so the dots still coincide red on red blue on blue lief which translations re ections and rotations leave the two dots xed Which translations re ections and rotations put the blue dot on top of the red dot and the red dot on top of the blue dot Now try this with three dots in of course three different colors which are not collineari Which translations re ections and rotations leave the three dots xed Now try this with a straight line Pretend that it s in nite Which translations re ections and rotations leave the line xed Which translations re ections and rotations don7t leave the line xed but still leave it lying on top of itself 2 The basics 21 Transformations Let7s formally de ne what a transformation is De nition 1 A transformation of a space S is a map 45 from S to itself which is 11 and onto Notation ab S A S Remember that 11 means that if p q are different points then f that is there7s no more than one way to get to any given point in 5 via while onto means that for every point 4 there is some point p such that L that is there7s at least one way to get to any given point via Here are some examples of transformations of R2 the plane 1 Re ecting the plane across a line 2 Rotating the plane about a point by a given angle 3 Translating a plane by a given vector 4 Contracting or expanding the plane about a point by a constant factor All of these kinds of transformations can be applied to R3 3 space as well with some modi cation For example re ection in R3 takes place across a plane not across a line and rotation occurs around a line not a point Question for those who have had some linear algebra or vector calculus How do these various transformations behave in R Here are some functions that are not transformations 1 The function taking all points zy E R2 to the point z E R It s not 11 and the space you start with isn7t the space you end up with1 2 The map taking all points z E R to the point 10 6 R2 Its not onto and the space you start with isn7t the space you end up with even though R is geometrically isomorphic to its image 3 Folding a plane across a line Z this is 21 rather than 11 off Z and it isn7t onto the whole plane 4 The function f R A R de ned by 12 It s neither 11 nor onto On the other hand the function 91 13 is a transformation An important note When we talk about transformations we only care about the map not how it is described For example the following three descriptions all describe the same transformation 1This is still an interesting map geometrically even though it isn7t a transformation Its an example of projection in this case projecting a plane onto a line o rotate the plane by 90 about the origin 0 rotate the plane by 72700 about the origin 0 re ect the plane across the zaxis then re ect across the line y I To be precise we consider two transformations 45 S A S and 1 S A S to be the same iff for all points p in 5 Re ections rotations and translations have a special property they don7t change the distance between any pair of points That is these transformations are isometries2 Welll come to back this idea later 22 Groups Transformational geometry has two aspects it is the study of transformations of geometric spaces and it studies geometry using transformations The rst thing people realized when they started to get interested in transformations in their own right in the 19th century was that there was an algebra associated with them Because of this the development of the study of transformations was closely bound up with the development of abstract algebra In particular people realized that transformations behaved a lot like numbers in the following ways 0 Closure Since transformations are functions you can compose any two transformations to get another transformation Speci cally if 45 and 1 are transformations of a space S then so is ab 0 Remember this means rst do 1b then do 415 ie an Mp we 0 Existence of an inverse Recall the de nition of an inverse function q5 1p q if 11 Since every transformation of S is 11 and onto it has an inverse which is also a transformation of 5 Remember that inverting a function switches its domain and range but in this case both domain and range are just 5 0 Existence of an identity element The identity transformation denoted id is the transformation that leaves everything alone idp p for all points p E S o Associativity If 45 1 and w are three transformations of a space then 45 o o w o o w These four properties show up together in a lot of places For instance consider the set R of real numbers and the operation of addition If you add two real numbers you get a real number Every real number has an additive inverse namely its negative Therels an additive identity namely 0 And addition is associative ab 0 a bc One way to think about associativity is that it doesn7t matter how you parenthesize an expression like a b 0 These properties together 7 we can compose two transformations to get a new transformation there is an identity transformation every transformation has an inverse and composition is associative 7 say that the transformations of a given space form an algebraic structure called a group Analogously the real numbers form a group because we can add two real numbers to get a real number there is an additive identity every real number has an additive inverse and addition is associative One big difference between the group of real numbers and the group of transformations is that addition is commutative but composition of transformations is not That is if 7 s are real numbers then T s 8 7 2From Greek iso same metry distance but if 4511 are transformations then it is rarely the case that ab 0 1b 1b a Thatls okay 7 the operation that makes a set into a group doesn7t have to be commutative but it does have to be associative The idea of a group is absolutely fundamental in mathematics3 As we7ll see later on groups come up all the time in geometry In some sense a lot of modern geometry is about groups just as much as it is about things like points and lines 23 Notation for transformations Here are the major types of transformations of the plane that welll study Transformation Notation Re ection across line Z w Rotation about point p by angle 9 pp Translation by vector 17 73917 Glide re ection rst re ect across line Z then translate by vector 17 71 Dilation about point p with constant factor k 617 In some sense these are the most interesting77 kinds of transformations though certainly not all possible transformations This notation makes it easier to describe relations between transformations For example the fact that re ecting about a line twice ends up doing nothing can be expressed by the following equation 77 o w id Instead of saying Rotating counterclockwise about a point p by angle 9 is the inverse transformation of rotating clockwise about p by 9 7 which is true but extremely awkward 7 we can write a simple equation 717071 ppr9 3 Transformations and geometry In the previous section we looked at transformations by themselves Now we look at the interaction between transformations and sets of points First one piece of notation If 45 S 7 S is a transformation of S and A is a subset of S then we7ll write A for the image of A under 5 That is MA 4510 l P E A If we read this notation symbol for symbol it says A is the set of all points 4510 where p is any point in A For example here7s the rst gure again 3To learn more about groups take Math 558 If A is the unshaded triangle and B is the shaded triangle then we can Write TAA B and TAB Al De nition 2 A transformation 45 xes a point p iff p lt xes a set A iff for all p E A p It is a symmetry of A iff 5 Example 1 Consider the following picture What happens to each of these lines under the re ection 7 1 0 First of all 7 1 xes every point on E itself So certainly TM Zr 0 Second TAa a On the other hand 77 does Lot x most of the points on a except for P it ips them across Z to other points that are also on a So w is a symmetry of a but does not x it 0 Third TAb c and TAG 12 So 77 is not a symmetry of b or of cl Some more brief examples to think about 1 pp xes p no matter What 9 is 2 If p E Z then pp is a symmetry of E but does not leave it xed 1300 3 If E is perpendicular to m then 7 1 is a symmetry of m but does not leave it xed On the other hand 7 1 does leave Z itself xed 4 lf17 0 then 73917 does not have any xed points On the other hand if E is parallel to 17 then 73917 is a symmetry of Z Using these terms we can rephrase the questions asked in Project 1 Which transformations x a single point two points three points a line Which transformations are symmetries of two points of a line First here7s an easy observation about the symmetries of all objects Fact 1 If 45 xes A then 45 is a symmetry of A In particular the identity transformation id of S xes every point in 5 so it is a symmetry of every subset of S 4 Special kinds of transformations isometries similarities and af ne maps The next step is to categorize transformations according to how much geometric structure they preserve For example consider the transformation of the plane that takes the point zy to the point Lyg letls call this transformation Note that 45 is 11 and onto as required On the other hand 45 is not very nice from a geometric standpoint For instance 45 takes the line y z and turns it into the curve y 13 So it doesn7t preserve straight lines And this means it doesn7t preserve the angle 180 so it doesn7t preserve angles lt doesn7t preserve distances either for example the points 11 and 12 are at distance 1 from each other but 45 sends them to 11 and 18 which are at distance 7 45 is an example of the kind of transformation we are not interested in De nition 3 Suppose we have a transformation l 45 is an isometry iff it preserves distances that is if A and B are any two points then AB A B where A and B 2 45 is a similarity iff it preserves angles that is if A B C are any three points then ABC 2 AA B C where A B and Ch 3 45 is an af ne map iff it preserves straight lines that is if X is a line then so is X and if is a line then so is X We7ve already observed that if 45 is any rotation re ection or translation then it is an isometry Therefore 45 is also a similarity and an af ne map Dilations are similarities and are af ne maps but not isometries In fact the ideas of isometry similarity and af ne map are successively more and more general Theorem 1 Every isometry is a similarity but not every similarity is an isometry Every similarity is a ne but not every a ne map is a similarity Proof Every isometry is a similarity by SSS Speci cally suppose 45 is an isometry lf A B and C then the triangles ABC and A B C are congruent by 555 so ABC 2 AA B C so 45 is a similarity On the other hand dilations are similarities but not isometriesi If 45 is a similarity then it preserves angles so in particular it preserves the angle 1800 That is it preserves straight lines On the other hand we7ve just seen in the exercise above a map that is af ne but not a similarity D Theorem 2 The isometries form a group the similarities form a larger group and the a he maps form a still larger group Proof Well just consider the case of isometries 7 the proofs that the other two sets are groups work exactly the same way To prove that the set of isometries forms a group we show that it satis es the four conditions listed in Section 22 1 Closure We need to show that the composition of two isometries is an isometry ie that if 45 and 1 preserve distance then so does iloqbi Let A B be any two points and let A 45A B B A A B B i Then AB A B because 45 is an isometry and A B A B because 1 is an isometry but that means that AB A B and A cab A and B o Therefore 1 0415 is an isometry by de nition 2 Inverses Suppose that 45 is an isometryi In particular 45 is a transformation so it has an inverse transformation 4V1 which we want to show is af ne So let A B be any two points and let A 1A Bquot 1Bi Then A and Bi Since 45 is an isometry AB ABi Thatls exactly what we need to show that qb l is an isometryi These were the hard parts 3 Identity element The identity transformation is an isometry because clearly AB idA idBi 4i Associativity lsometries are functions so their composition satis es the associative lawi D Not every class of transformations forms a group For example consider the set of all re ections 7 1 l E is some line This set only satis es some of these properties so it does not form a group For example the composition of two re ections is not a re ection Another example For every point A the set of transformations 45 such that A forms a group However if A and B are different points then the set of transformations such that B does Lot form a group 5 The structure of isometries In this section we focus on isometriesi There are three major theorems about isometries Two of their proofs are fairly complicated so we won t give themi But we will give applications The ThreePoint Theorem Every isometry of the plane is determined by what it does to any three non collinear pointsi That is 4511 are isometries and A B C are noncollinear points such that A MB B7 and 4150 C7 th lt15 ill The proof of this theorem is rather technical but you7ve already seen the idea behind it 7 think about the threedot example in Project 1 The ThreePoint Theorem is useful for checking whether two isometries are equal all you have to do is check that they agree on each of three noncollinear pointsi Of course you may have to use some ingenuity in choosing those points appropriately Example 2 Suppose that mn are perpendicular lines that meet at a point A see gure below We will prove that Tm o Tn pAJgOOi We need to nd three noncollinear points and describe what each of these two isometries 7 the composition of re ections Tm o Tn and the rotation pAJgOO 7 does to themi The point A is a clear choice for one of the three points For the others let s draw a square BCDE centered at A with its diagonals parallel to m and n shown in blue below Why Because all the transformations we7ve described are symmetries of this square so its easy to see what they do to its verticesi B C 7 E D We see that MM A7 TAB D7 TAO 07 MM A7 MAD D7 MAC E7 and therefore Tm o TnA A Tm o TnB D Tm o TnC E On the other hand pA180 A A7 pA180 B D7 pA180 C E So the three noncollinear points AB C are mapped to the same pointsinamely ADE respectivelyiby Tm on and pA 1800 Therefore by the ThreePoint Theorem Tm urn pA 1800 which is what we were trying to prove D The ThreeRe ection Theorem Every isometry is the composition of at most three re ections If you believe the ThreePoint Theorem then you can prove the ThreeRe ection Theorem constructively and in fact you will do so as a homework problem That is if 1 is any isometry then the ThreePoint Theorem says that 1 is de ned by what it does to any three noncollinear points ABCi So to prove the ThreeRe ection Theorem it is sufficient to show that if AB CABC are six points such that AABC AA B C then there is some way of transforming AABC to AABC using three or fewer re ectionsi One application of the ThreeRe ection Theorem is the following theorem 7 which in case you thought everything was about the number 3 is about the number 4 The Isometry Classi cation Theorem There are only four different kinds of isometries re ection rotation translation and glide re ectioni What about the identity it can be described as either translation by the zero vector or as rotation about any point by 00 There are many ways to prove this theorem all of them tedious so we won t give a proof But all of the proofs rely to some extent on the ThreeRe ection Theoremi On the other hand the lsometry Classification Theorem has a nice corollaryi Theorem 3 Let 45 be an lsometry Either 45 is a symmetry of same line or it xes a point Proof By the lsometry Classification Theorem there are only four cases to consider o if 45 is a re ection 7 1 then 45 fixes every point on E so it certainly is a symmetry of Zr o if 45 is a rotation pp 9 then it xes the point p o if 45 is a translation 7395 then it is a symmetry of any line parallel to the translation vector 17 o If 45 is a glide re ection Wm then it is a symmetry of the line if D How many symmetries does a regular tetrahedron have How about a cube Or an icosahedron 6 Symmetries of bounded gures What can we say about the set of symmetries of a figure4 First of all let s agree that when we talk about a symmetry of a figure we restrict ourselves to isometriesi This captures our intuitioni There are many transformations that look like isometries in a small region of space but then do strange things outside it and it complicates our discussion too much to talk about those Call the figure yr The symmetries of y are closed under composition the identity transformation of the plane is a symmetry of y and each symmetry of g has an inverse which is also a symmetry of yr So they form a group which we7ll call Sym i 61 An example Suppose that y is an equilateral triangle AABCi in addition to the identity the group SymAABC contains two notrivial rotations 7 namely pZ 120 and pZ 240 where Z is the center of the triangle 7 and three re ections 7 1 Tm and Tn where Lm nare the bisectors of the three sides of the triangle 4Here 17m using the Euclidean de nition of gure an object built out of curves and line segments So triangles pentagons and circles are gures but not for example a lledein circle That is SymAABC id palm pzjm 7 1 Tm Tni Here s how we know that this is the complete list of symmetries Every symmetry 45 of AABC takes vertices to vertices that is 45 is a symmetry of the set ABCi For example Tm xes B and swaps A with C while pZ 120 maps A to C B to A and C to B The identity of course xes each of the three verticesi On the other hand by the ThreePoint Theorem any isometry is determined by what it does to A B and Cl So there are only 3 6 possibilities which means that we7ve listed them all Since the symmetries form a group we can ask how they behave under compositioni That is if 4511 are transformations in SymAABC then which element of SymAABC equals ab 0 1b This question is really a set of thirtysix questions eigi What is palm o Tm What is Tn o T73 whose answers can be collected in a table The easiest way to calculate a single composition is to see what it does to AB C For example pz120 MUD pz120A C TmA WTmA TAO B Pzgm Ab pz120WB pz1200 B TmB WTmB TAB C Pz240B7 pmww pmw A we wltrmltcgtgt MA A who so palm a 7 1 Tm and 7 1 0 Tm r2124 In the following table the rows and columns are labeled by the elements of SymAABC and the entry in column 45 and row 1 is ab 0 id pz120 pz240 W Tm T id id pz120 pz240 W Tm T pz120 pz120 pz240 id Tm T W pz240 pz240 id pz120 T W Tm T2 T2 Tm Tn id P2420 1240 Tm Tm M W 1240 id P2420 Tn Tn W Tm pz120 pz240 id Notice the following things 10 o It is not always true that ab 0 1 9 1 o For example if r1 0 rm but rm 0 7 1 pz2407 pZ120 o The composition of two rotations or of two re ections is a rotation while the composition of a rotation and a re ection in either order is a re ection This is analogous to the sign of the product of two real numbers the product of two positive numbers or of two negative numbers is positive while the product of a positive number with a negative number is negative 62 De ning gures by their symmetry groups To a modern geometer ie any geometer since the late 19th century what characterizes a geometric gure isn7t the number or characteristics of its sides andor angles but its symmetry group For example suppose that y is a convex quadrilateralr Most of the time g has only one symmetry namely the identity transformation But we can identify several special kinds of quadrilateralsr SOQAQ isosceles square rectangle rhombus parallelogram trapezoid kite What makes these special quadrilaterals special is exactly that they have a lot of symmetriesr In fact we can de ne them in terms of their symmetry groups A parallelogram is a convex quadrilateral whose only nontrivial5 symmetry is palm for some point C which we call the center of the parallelogram it s where the diagonals meetl A rectangle is a convex quadrilateral with one nontrivial rotational symmetry and two re ection symmetriesr So is a rhombusl An isosceles trapezoid has just the identity and one re ection symmetryr A square of course has the most symmetries of any quadrilateral four rotational symmetries including the identity and four re ection symmetriesr By the way we could de ne a circle as follows Let p be a point A circle with center p is a gure y such that every rotation around p is a symmetry of y and every re ection across a line containing p is a symmetry of ya This seems like a roundabout way to de ne a circle but if you think about it its correct 7 every circle certainly has these symmetries and any gure with these symmetries has to be a circle A gure is called bounded if it ts inside some circlel For example a triangle is bounded a line isnltr Not all isometries occur as symmetries of bounded gures Theorem 4 If y is a bounded gure in the plane then every symmetry of y is either the identity a rotation or a re ection Equivalently no translation or glide re ection can possibly be a symmetry of y The equivalence of these two statements comes from the lsometry Classi cation Theoremr Here s a really slick proof that depends on the de nition of circle 5By nontrivial we mean other than the identity 11 Proof If y is bounded then there is some unique smallest circle 9 that y ts inside Every symmetry of y must be a symmetry of if and by the de nition of circle must be either a re ection or a rotation D In particular this means that every symmetry of y xes at least one point namely the center c of 9 We could call this point the center of y For example if y is a triangle then if is the circumscribed circle so c is the circumcenter of L that is the intersection of the perpendicular bisectors of the sides 6 3 Regular polygons Most gures don7t have any nontrivial symmetries For example if you draw a random triangle then it will almost certainly be scalene and the only isometry that xes it will be the identity However there are special gures with more symmetries The equilateral triangle of Section 61 is an example of this More generally De nition 4 A polygon P is regular if all of its sides are congruent and all of its angles are congruent Suppose p is a regular nsided polygon or ngon How big is the set SymP First of all let c be the center of P If 45 E SymP then c Also 45 takes vertices to vertices and it preserves adjacency among vertices if v and w are vertices of P that are adjacent to each other then so are and But the points c 1110 are noncollinear 7 but by the ThreePoint Theorem 45 is determined by what it does to all of them There are clearly n possibilities for namely all the vertices of P and once we know 451 there are 2 possibilities for namely the vertices adjacent to 451 so we conclude that P has exactly 2n symmetries In fact it is not too hard to say what the symmetries are Theorem 5 Let P be a regular polygon with n sides The nontrivial symmetries of P are as follows 0 all reflections across its angle bisectors 0 all reflections across the perpendicular bisectors of its sides and 0 all rotations about its center by 360hn for 0 lt h lt n Proof All of these transformations are certainly symmetries of P On the other hand if 45 E SymP then 45c c where c is the center of P as de ned above so 45 must either be a rotation about p or a re ection across a line containing c and any rotation or not re ection that is not one of those listed above does not take vertices to vertices D 64 Other polygons What about polygons that are not regular but have lots of symmetries nevertheless For example what does the group of symmetries of a rectangle look like Remember we said that a rectangle is a convex quadrilateral with one nontrivial rotational symmetry and two re ection symmetries across the perpendicular bisectors of each pair of opposite sides Of course so is a rhombus 7 although in this case the lines of re ection symmetry are the diagonals 12 Here s the multiplication table for the symmetry group of a rectangle id pX180 T1 Tm id id prgo 77 Tm pX180 pX180 id Tm W m w Tm id prgo Tm Tm W pX180 id And here7s the multiplication table for the symmetry group of a rhombus id py 180 Tk Tn id id py 180 Tk Tn pY180 pY180 id Tn T16 m m Tn id py 180 Tn Tn m py 180 id These two multiplication tables are essentially the same if you take the rst table and replace X with Y7 E with k and m with n7 you get the second tablei Algebraically7 we say that the symmetry groups of the rectangle and the rhombus are isomorphic Therels a good reason for this the two gures can be superimposed so that their symmetry groups consist of exactly the same sets of transformations 13 This is an example of how modern mathematics uses groups to study geometric objects The fact that the symmetry groups of the rhombus and rectangle are the same indicates that there s some relationship between the two gures Of course you don7t need groups to realize that you can form a rhombus by joining the midpoints of a rectangle but the same technique can be applied to more complicated gures 7 Counting symmetries If we know the symmetry group of an object that is if we know its multiplication table then of course we know how many symmetries there are But it is often possible to count the symmetries without having to work out the full symmetry group Example 3 Let 73 ABCDE be a regular pentagoni Every symmetry 45 of 73 permutes its ve ver tices that is it is a symmetry of the vepoint set AB CDE and by the ThreePoint Theorem is completely determined by what it does to any three of the ve But actually 45 is determined by even less information For instance if we know what and are then we know 45 completely by the ThreePoint Theorem again 7 because 0 where O is the center of 73 and the points A B O are noncollineari The point can be any of the 5 vertices of 73 and once we know 45A we know that must be one of the 2 vertices sharing a side with whatever that is Therefore the number of symmetries of 73 is 5 2 10 We can describe a symmetry of by its permutation wordi That is write the ve letters A B C D E in the order 45A 4MB i i i Here are the permutation words for all ten symmetries of the regular pentagon ABCDE AEDCB BAEDC BCDEA CBAED CDEAB DCBAE DEABC EABCD EDCBAA For example p0gt144o correponds to the permutation word DEABC because p0144o A D p0144o B E etc and Tb corresponds to the permutation word CBAED because TbC A TbB B etc The permutation word ABCDE corresponds to the identity transformation Notice that each of the 5 possible rst letters occurs twice in the table once with each of its neighbors next to it This corresponds exactly to our earlier observationi Generalizing this argument we can see that every regular polygon with n sides has exactly 2n symmetriesi Of course we already knew that from Theorem 5 but its nice to con rm it another way This method of counting symmetries doesn7t tell us explicitly what the symmetries are but on the other hand it is applicable to lots and lots of geometric objects 7 not just in the plane but also in threedimensional space and even in four and higherdimensional spaces 14 Example 4 Let R WYXZ be a rectangle that is not a square as shown below and let 45 be a symmetry of R Then can be any of the four vertices but once we know 45X there s only one possibility for 45 because must be the vertex adjacent to by one of the short sides of Qi HQ were a square then there would be two choices for instead of one So Sym 4 which con rms what we found earlier The permutation words for the four symmetries are WXYZ XWZY YZWX ZYXW Similarly let H ABCDEF be the hexagon you studied in homework problem TG 15 and let 1 be a symmetry of Hi Again can be any of the six vertices but once you choose A you immediately know what 1 does to the other ve vertices of Hi Therefore SymR 6i There7s nothing special about A we could just have well argued that 1 is determined by which can be any of the six verticesi What about higherdimensional objects Example 5 Let T be a regular tetrahedron ie a triangular pyramid in which every side is an equilateral triangle Call the vertices AB C Di How many symmetries does T have Equivalently what are all the permutation words of symmetries of T If 45 is a symmetry then clearly can be any of 45A 4MB or Four choices there 15 Having chosen 4514 there are three choices for any of the other three verticesi Having chosen and 4MB there are two choices for Then once we choose 41507 there is only one possibility left for of the other three verticesi ln total7 there are 4321 4 24 symmetries of Ti ln fact7 every rearrangement of the letters A7 B7 C7 D is a permutation word of a symmetry What these symmetries look like geometrically For example7 you can draw a line connecting a vertex with the center of the opposite triangle and rotate T by 120 or 240 around this line7 as in the following gure There are four ways to choose that vertextriangle pair7 so we get a total of eight rotations this way Here are the permutation wordsi Vertex Opposite triangle Permutation words A B C D BOD ACD ABD ABC ACDB7 ADBC CBDA7 DBAC BDCA7 DACB BCAD7 CABD Another way to construct a rotation line is to connect the midpoints of two opposite edges of T as in the following gure ltls probably easiest to visualize if you dangle T from one of its edges 7 the righthand gure is an attempt at illustrating this 16 There are three such pairs of opposite edges AB and CD AC and BD and AD and BCi This gives three more permutation words7 respectively BACD7 CDle7 and DCBAi We7ve accounted for twelve symmetries so far the identity and 3 8 11 nontrivial rotationsi The other twelve are re ections for example7 re ecting across the plane containing edge AC and the midpoint of edge BD or compositions of re ections and rotationsi 17

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