1. Glasser
Rhodes University Grahamstown, South Africa
I I
Teaching Symmetry The use of decorations
Symmetry is a concept which receives considerable attention in the current literature because of its importance in the study of crystalline structure and molecular structure (spectroscopy, ligand field theory, molecular orbital theory, etc.) (1); the essentials of the concepts of symmetry are embodied in the elegant theory of the representation of groups (2). To understand and apply the symmetry properties a ready mental picture of the symmetry operations and groups is vital, as is the ability to detect symmetry elements within a given molecule or structure. Being an essentially geometrical concept, symmetry is especially suitable for study by means of diagrams and models, the use of which can enhance that ability to visualize in three dimensions which is often lacking in chemists but is easily developed, as it is by engineers who are more used to dealing with concrete objects through the medium of diagrams. A particularly direct form of presentation of symmetry is by the use of decorations, notably the drawings of M. C. Escher,' which can be used to illustrate the various kinds of symmetry and their combinations to great advantage, for "these patterns are complicated enough to illustrate clearly the basic concepts of . . . symmetry, which are so often obscured in the clumsy arrays of little circles, pretending to be atoms, drawn on blackboards" (3). These illustrations are confined to two dimensions (although sometimes depicting objects occupying space), and need to be supplemented with solid models, but the ideas developed with their aid are readily extended to three dimensions, the mind's eye having been prepared for this extension. There is a further important benefit, for the unusual nature of the decorations introduces a dramatic impact into the discussions and can inspire a special interest in symmetry in the student. The decorations are readily presented with the aid of an overhead projector where the transparent prints can be turned, drawn on, over-
' Maurits Escher, a. Dutch graphic artist, has been deeply concerned with attempts to render on paper the appearances of fonns in space and this concern, complemented by his mastery of graphic techniques, has led him to many strange and surrealistic coucep tions. Apart from the essentially two-dimensional decorations presented here, he bas a large oeuwe of both representational works and of works concerned with perspective and spatial modeling. His vorks have appertred in a. number of scientific publications; a list of these is given below: Reference (3) of this article cited in ref. (6) of this article Paper by P. TERPSTRA M., Scientific American, 214 (April), 110 (1966), GARDNER, and references cited therein TERPSTRI,P., AND CODD,L. W., "Crystallometry,"Longmans, London, 1961 2 This fixed point could be any point,-inside or outside the hody, in the trivial operation of rotation through 360°, symbolized 1 or
c,.
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laid with lattice networks or symmetry symbols, etc.
(4).
Escher's drawings (5) have been successfully applied to the presentation of point symmetry by the present author for a number of years, the original stimulus in the use of decorations being Weyl's imaginative book (6). While the translational aspects were not neglected, it has been possible to extend their scope as a result of the availability of extra material (3). A further, extraordinarily rich, source of decorations is "The Grammar of Ornament" (7), from which some decorations (Fig. 5) are also reproduced in this article. While a number of the drawings discussed here are well-known to crystdlogmphrrs, they are unfmdiar to rhe wider uudielw of sviel~risrs.r l ~ it d is a ourrwsr of the present article to make a few of them known to teachers who may be stimulated to use such drawings in their own teaching. A
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Point Symmetry
Crystals and molecules, and, in fact, all finite bodies possess more or less symmetry depending partly on the condition in which we find the bodies, the detail with which we examine them, and the geometrical idealization we permit. The possession of "symmetry" by such a hody means that if the appropriate geometrical (such as rotation) or hypothetical geometrical (such as reflection) operation is performed on that body, it will be reproduced exactly in its original position in space. This must imply that at least one point in the body, its center of gravity, is unmoved by the geometrical o p e r a t i ~ n . ~Symmetry of this kind is termed point symmetry (to distinguish it from translational or space symmetry). There are two essentially different operatims of point symmetry, termed proper and improper (or reflexive) rotations respectively, each operation being associated with a symmetry e l a a t in the symmetrical body being examined; the operations describe the processes to be performed with I'espect to the elements. Proper Rotations
The simplest type of symmetry is proper rotational symmetry (Fig. 1). The pattern in the figure may be rotated in its own plane about a vertical axis through its center and repeats itself exactly after (360°/2) = 180'. The axis about which the rotation is performed is termed a 2-fold (rotation) axis. More generally, the fold, or order, of an axis when a body is repeated after rotation through (360°/n) is symbolized by n (in the Hermann-Mauguin notation favored by crystallographers) or by C . (in the Schoenflies notation
Figure 2. Cube (point group m3m1, with 1.1- an octahedron (point group m3ml mnd (bl o tetrahedron Ipoint group 43m) ihwribed. In (a) one of the 4-fold ores and o mirror plone is shown, while in (b) the 4-fold oxir has degenerated to o ?-fold axis. The four 3-fold axes chorocterirtic of cubic symmetry are also rhown. Other point symmetry e l e m e n h in there figures are not shown. (c) The graphical symbols for the point symmetry elemen& are rhown here (I 11.
Figure 1.
Two-fold rototionol rymmetry.
favored by theoreticians and spectroscopists). The rotat,irm through 360°/2 is the only symmetry in this pattern; this fact is reflected in the behavior of the fish which themselves rotate, swimming head-to-tail in circles. The restriction of the figure to two dimensions is represeutdtive of the examination of a body or molecule in projection; all bodies which contain proper rotation axes must show those axes in the appropriate projection. The converse need not hold, for bodies which show proper rotation axes in project,ion may have only improper rotation axes in reality. If the shade differences of the fish in I'igure 1 are ignored, then the vertical axis is 4-fold. An axis of order 2n always contains, as well, one of order n (in this case, the 4-fold axis contains the 2-fold axis discussed above). Ignoring the shade difference has a very real analogy in the practice of crystal structure analysis for, whereasx-rays are insensitive to magnetic effects when being scattered from atoms, neut,rons can be vex>- differently scattered by isotopes of different magnetic moment (S), so that the symmetry of a crystal structure observed using X-rays can become reduced when using neutrons. This kind.of distinction has been incorporated into symmetry theory by the development of black-white and color (polychromatic) symmetry theory (9) and is well represented by diagrams in reference (3). An important group of bodies which ran (but do not necessarily have to) contain 4-fold axes are those bodies which have cubic symmetry (Fig. 2), such as the cube itself, the octahedron, or the tetrahedron, whose symmetries are adopted for example by the molecules cubane (C8Hs), SFB, and P,, respectively. It can be seen in Figure 2 that the 4-fold axis of the cube degenerates to a 2-fold axis in the tetrahedron. The characteristic symmetrj- of cubic bodies is the set of four 3-fold axes lying along the body-diagonals of the cube, andinclined a t 54' 44' t,o axes through the centers of opposite faces of the cube.
Rotation axes of order from 1 to m are found in molecules. A linear molecule, like Con, contains an infinite-order axis along its length while spherical bodies, such as rare gas atoms, have an infinite number of infinite-fold axes passing through their centers of gravity. Examples of simple molecules containing rotational axes as their only elements of symmetry are : CHGHO
(0
Hz03 (2)
HCI
(-1
improper Rotations
The second class of symmetry operations is that of improper rotations which, as well as rotating, convert left into right and vice versa, i.e., they generate nonsuperimposahle replicas (enantiomorphs) of the body, and the space, operated upon. This sort of operation cannot be performed by a simple geometrical transposition, but requires additionally the conceptual processes of reflection or inversion, which we now discuss. The creature in Figure 3 has mirror or bilateral symmetry, all parts to the left being equivalent to parts to the right by reflection in a plane mirror passing vertically along the length of the creature. If the
Figure 3. T h e "curl up" portrayed has a vertical pione of rymmctry only. This creature has the unique feature of using t h e wheel 01 a mode of locomotion.
mirror is taken to define the yz plane of a set of Cartesian coordinate axes, then the action of the mirror is to reproduce all parts with coordinates x, y, z a t -x, y, z. The operation is termed (mirror) reflection or inversion across a plane, and the symmetry element is given the special symbol o in spectroscopic notation and m in crystallographic notation. The operation of inversion through a center (the center of symmetry) consists in replacing each point of the body, having coordinates x, y, z, with an identical Volume 44, Number 9, September 1967
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point an equal distance from the center but on its opposite side; the coordinates of the neu7 point are -x, -y, -2. Figure 1, our example of a 2-fold rotation axis, displays this symmetry, since a center of symmetry degenerates to a 2-fold axis in two dimensions. The center of symmetry is given the special symbol i in spectroscopic notation, but has no sperial symbol in crystallographic notat,ion. Inversion across a plane and through a ceuter are but special cases of the more general operation of improper rotation. T o describe the improper rotation, the Hermann-Mauguin notation system favors a composite rotation-inversion operation while the Schoenflies system uses rotat,ion-reflection. For the rotation-inversion operation, imagine the creature of Figure 3 impaled on a pin through its side, rotated through (360/2) = 180°, and then inverted through the center of the rotation axis. Then t,he former left side becomes the right, and vice versa, and yet the creature is reproduced in its original orientation exartly. This combined operation (2-fold rotation inversion) is termed a 2-fold (rotary) inversion (symbolized 2 in the Hermann-Mauguin notation) and is, as we now see, identical to a mirror reflection (112). It is essential t,o note that the intermediate situation, after rot,ation hut before inversion, does not reproduce the original situation and so rotation (or, for that matter, inversion) does not constitute part of the symmetry on its own, as can also he seen by an examination of Figure 3. Exactly the same operation may he descrihed by rot.ation combined n-ith refledon, that is rotation through (360°/1) = 360' about the axis through the creature's side, combined with reflection across the plane normal to the rotation axis, the operation being described as 1-fold rot,ation-reflection, the axis as a rotat,ion-reflection or alt,ernating axis (symbolized 1 in Hermann-Mauguin not,ation, S, in Schoenflies notation). The equivalenre of the two operatior~sand of the various notatious may he written compact,ly in symbols:
+
Hermann-Mauguin n-here, tno, the simple mirror reflection is symbolized 111 rather than 2. I n two dimensions, improper axes degenerate t o proper axes of halved, equal, or doubled order. 111 three dimeusions they are found, for instauce, in the tetrahedron (Fig. 2b), vhere the 2-fold axis is more completely described as a. ?-axis. This description accounts for the combined symbol for 4- and 2-fold axes shown along this axis in Figure 2b. A complete set of the conventional symbols for elements of point symmetry is shown in Kgure 2c. An example of a simple molecule containing only an improper axis is HOD@ = In, being the molecular plane). Combinations of Point Symmetry Elements
The point symmetry elemeuts may he grouped in n number of urays, ench con~binationbeing called a point group. For instance, a diagram in which a 3-fold axis is combined with a mirror plane parallel to it is shown in Figure 4. Of course the &fold axis not only rotates the pattern hut also the symmetry elements, so that there are three vertical mirror planes, in all, in the figure. Au infinite number of point groups are
u
It should uow be clear that rotation-inversion and rotation-reflertion operations can lead to the identical results, though the orders of the rotation axes will not, in general, he the same in the two operations. Table 1 shows the relationship between the axes of various order. I t is conventional t o use rotation-reflection axes in the Schoenflies notation but rotation-inversion axes in the
Table 1.
The Equivolence of the Various Rototion-Reflection ond Rotation-Inversion Axes
Symhol for rotatiow inversion
Special srmhol
Symbol for rotationreflection
The wder oi t,he two types of improper axes is the same when the order of the axis is divisible by 4. Symbols llot nmally used for symmetry elements are enclosed in parentheses.
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Figure 4. A 3-fold symmetry oxir parollel to a mirror plane, point group 3 m (E Caul.
possible (because of the infinite number of rotation axes), but if we limit the axes t o those which are compatible with translation, viz., 1, 2, 3, 4, and 6 (this limitation is discussed below), then there are only 32 diierent point groups possible; these are called the crystallographic point groups. Symbols for these point groups are formed by juxtaposing in a conventional fashion the symbols for the various elements of symmetry contained in the group, with the symbol for the principal axis (the axis of highest order) being placed first. The symbols of symmet,ry elements generated by combination of symmetry elements are, in general, omitted as redundant. Thus, tu-o mirror planes perpendicular to one another necessarily hare a 2-fold axis along their line of intersection: in the Hermann-Mauguin notation this symmetry is described
as mm, omittirig mention of the &fold axis, while in the Schoenflies system it is C2.-a vertical 2-fold axis (the principal axis is always considered to be vertical in Schoenflies notation), together with a vertical mirror plane (v)-omitting mention of the second mirror plane. Table 2 lists the various symmetry element combinations and their symbols. Listings of the point groups and diagrams of their symmetries appearin references (10) and (11). Table 2. Symbolism for Various Combinations of Symmetry Elements.
Ccmbinntion Rotation axis Inversion axis Axis tilth mirror plane normal to it Axis with 2-fold axes normal to it Aris w t h mirror planes uaralld t,o it Axis with a mirror plane normal t,o it,, and minnl. ulanrr onrallel to it
Hemam-Mauguin Svmhol
n = n/m n~
n2
n rn
n m = n/mm I ~ L
The use of Schoeatlies is not poaiihle in this context, as they refer to complet,e point groupa ~.ntherthan to isolated cambinsi tions of element,s. Translation Symmetries3 One Dimension (Line and Band)
We have confined ourselves to symmetry about a point, but t,his is not the only symmetry possible. It is possible to take some object arrd repeat it by translation along a given direction, as in Figure 5a. The translation vector measures the distance from a given point to the next point which is equivalent by translation alone. Translation can be combined with a transverse reflection plane as in Figure 5b, where it can be seen t,hat there are two different kinds of mirror plane, mutual1~-spaced a t one-half a translation distance. Such a pairing of mirror planes must necessarily occur, for each mirror plane of one type has a left and right image on either side, and so right aud left image pairs must lie between the mirror planes, where they are related by additional mirror planes bisecting t,he translation. These are the only possible symmetn combinations within a h e . Figure 5c shows a translation combined with reflection across the translation. This reflection is really independent of the translation, however, because each individual part of the pattern has its enantiomorph nith it a t the same point in the translation; together, this pair oonst,itutes the patteru upon which the translation operates. Such a pattern with a finite transverse dimension and infinite longitudinal ex-
Only the Rermmn-Mmgnin natation will be used hereafter because it, provides s complete repre=ntat,ion of the space sym-
metry; Schoenflies notation has been used ta designate space groups but it is based on the point symmetry alone, the spaoe groups being placed in an arbitrary numerical sequencesee ~IUDMAN, R., J. CHEM.EDUC.,43, 682 (1966). It is to he noted that in this section we will discuss plane pa& terns with mirror lines in them; these lines will he called mirror "planes" implying that they represent the traces of planes passing normally through t,he plsne patterns under consideration.
Figure 5. lo1 A Chinese decoration containing two motifs in the pattern. repeoted b y pure trandolion in one dimension. I b l A Greek decorotion having translotion combined with reflection. There are two sets of mirror lines tronrver$e to the tran~lotion. Ic) A decoration from the Middle Ages having translotion combined with reflection mcros the tronrlation. ( d l An Indian decorationwith o glide tronrlotion.
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tension is a band ornament. Figure 5d is more complicated. Here me note that there is a reflectiori across the translation, but that the reflerted pattern is itself carried forward with the translation. The translation involved in a single reflection is not the true translation which, by definition, carries the complete pattern forward into an identical representation of itself. I t is only after a second step (translation reflection) that the true translation is completed. This combination operation (translation reflection) is termed a glide (g, in one dimensional symmetry symbolism), and has an important place in crystalline symmetry. The glide translation is necessarily half of a true translation in one dimension. There are, in all, only seven band ornamerrt symmetries, and all were known to the Ancients. We have examiried four longitudinal reof these: translation, translatiou flection plane, translation transverse reflection plane, and simple glide reflection plane (10, Phillips).
+
+
+
+
Two Dimensions (Plane)
A second, non-parallel, translation vector may be added to the first to generate a pattern which fills a plane. Figures 6 and 7 are examples of this situation, where there are no kinds of symmetry other than translation. Equivalent points in the patteru may be marked and the collection of points thus generated constitutes a (plane) lattice. Any one of these points may be chosen as an origin, arid two lines drawn from this origiu to the nearest adjacent points. The parallelogram defined by these two lines is described as the unit cell of the lattice. If the unit cell of Figure 6 is Volume 44, Number 9, September 1967
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Figure 6.
A plane rymmetry pattern, plone g r o v p p l
moved around on the pattern it is seen to he of such a size as to cont,ain only one fish and one ship, i.e., the minimum to fully describe the pattern which is superimposed and repeated on the lattice. Such a cell of minimum size is described as primitive, and the plane symmetry group symbol for Figure 6 is pl, the p standing for primitive (and in lower case for a plane lattice), the 1 for a lack of any symmetry other than translation. (A centered cell, containing more than
Figure 8.
A plone rymmetry pottern, plone grovppmm.
Symmetry Combinations with Translation
Figure 9 introduces the glide line in a plane pattern (also symbolized g). I t will be observed that the light, (and the dark) figures can be divided into two groups, one with left arm fully visible, the other with right arm visiblethere is clearly a mirrur, or left-right, relationship between them. If a vertiral line is drawn hetween the backs of the calves of the light and dark figures it, will he seen that the pattern is mirrored on the t ~ m sides of this line, except for a translation of the height of a figure. This is the glide. It will be recalled that, in the case of a one-dimensional translation, the introductiou of a reflection along the translation illvolved the appearance of mirror planes of two different sorts. In the same way, the vertical glide (a kind of mirror plane) already picked out in Figure 9, combined xvith the translation sideways, introduces another glide with
Figure 7. A plone symmetry pattern, plone g m u p p l , since white and block are not interconvertible by translation or retlection.
one pattern unit, is sometimes chosen rather than a primitive cell in order that the unit cell may display the full symmetry of the lattice). For Figure 7, if we acknowledge that no geometrical operation can convert a black to a white knight, and vice versa, there is also no symmetry other than translation, and so the symmetry is also pl. Figure 8 is a little more complex. Here, combined with translation, there is mirror symmetry with traces of two sets of mirror planes cutting diagonally across the diagram, their intersections being %fold rotations, as mentioned before; the group symbol for Figure S is prim. 506
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Figure 9.
A planesymmetry pattern, plane grouppg.
different surroundings, passing vertically through the elbows of the men. The pattern has symmetry pg. With properly chosen unit cells, in three dimensions, glides may occur parallel to any of the three axial directions (symbolized a, b, or c glides), along a facediagonal of a cell, the glide operating over either half the length of the diagonal (an n, or diagonal, glide), or over onequarter of the diagonal's length (a d, or diamond, glide), or, sometimes, over one-quarter of a body diagonal (also d). It is appropriate to reconsider Figure 7, now neglecting the color difference between the knights, and noting only the similarities. I t is readily observed that the dark and light knights are mirror-images, and that there are two sets of vertical glide lines in the pattern, one set passing approximately through the hind-legs of the horses while the other set, halfway between them (cf., the two pairs of mirror planes in Fig. 5b), passes through the knights' ankles. This pattern then has the symmetry pg also. The second and final type of combined translational symmetry operation is that of translation together with rotation, termed screw rotation. The symmetry of a spiral (actually, helicoidal) staircase is described by a screw axis. The axis is symbolized n, where n is the order (just as for a rotation axis), the conventional screw rotation direction is taken to be that of a righthand screw (turning clockwise involves translation in the positive direction) and p/n is the fractional translation involved in each operation of the screw. Thus a 31 axis combines a rotation through (360°/3) = 120' with a translation of of a unit cell edge; after 3 operations the rotation is 360°, the translation is one unit cell length, and so the process continues. Left- and right-hand screws are readily distinguished by this notation. Consider one operation of a 3~ axis, which will involve clockwise rotation through 120°, together with translation through 2/3 of a unit translation; after a 2/8 translation (two operations) the 3, axis will have produced clockwise rotation through 240' = -120°. That is, the effect of the two axes is a similar translation together with rotation in opposite senses. The same holds for the pairs 61 and 6 ~62 , and 6&,41 and 43,etc. Table 3.
~
w
~
Screw axes cannot be conveniently depicted in two dimensions. If, in Figure 9, each man had his like bands similarly placed, then the men would be related by translation plus rotation about axes in the plane of the diagram and lying in the glide directions (if we consider the diagram to have an obverse and a reverse). The glides would then have become 2, screw axes. Symmetry i n Three Dimensions
If, to the two translation vectors of a plane lattice, there is added a third, non-coplanar, translation then a three dimensional or space lattice is generated. There are 14 essentially diiereot sorts of lattices, called the Bravais lattices, differing in their symmetries (see Table 3) and cell centering. An upper case letter is used in the symmetry symbol to indicate t,he centering of the lattice upon which the translation repetition of a crystal structure is based, as follows: P (= primitive, with lattice points at cell corners only); I (= body, or inner, centered); A , B, C (= one face centered); F (= all faces centered); R (= rhombohedral). If motifs, or pattern units, with their own particular point symmetries are superimposed upon the lattices, to create structures, there ~villbe many possibilities of symmetry combination; in fact there are 230 space groups (symmetry combinations, including translation, in three dimensions), as opposed to the 17 planegroups, a few representations of which were shown in Figures &lo. On the basis of the possession of certain characteristic symmetry elements, notably certain principal axes, the structures of crystals may he classified into 6-crystal systems, as shown in Table 3. The space group notation for a particular symmetry is constructed as follows: first, the upper case letter d e noting the Bravais lattice centering, then a set of the Hermann-Mauguin symbols showing the nature of the axes (rotation, inversion, or screw) their order being conventional (Table 3). Thus, P1g1 is the space group symbol for a monoclinic (because of the 2/m symmetry) lattice containing 2-fold axes normal to mirror planes, all repeated at appropriate intervals in space by the translations, and referred to a primitive unit cell. The 2 / m symbol in the second position indicates that
Characteristic Symmetries of the Crystal Systems and Conventional Order of the Symbols in the Point and Full S ~ a c eG r o u ~Svmbols ( 1 1 )
~2, m, d 2i / n~
i
~
2 and/or 5, along the unique axis ( n is equivalent to t ~ o mof the improper ans)
2 or 2, in one direction only
5, b u t
normal to the diree.
Orthorhombio 222, m"'2, mmrn
Three 2's and/or
0-*xis
I's
2 and/or 2
&axis 2 and/or
Te_tramnal 4 . 1 . 4/m; 422. 4mm. 4zm, 4 1 ~ ~ ~
4or1
c-axis 4 and/or 1
a = b axes 2 and/or i
2 and/or 2
Trigon31 and Hexagonal 3 . 3 : 32. 3". 3m; 6. 6. 61m; 622. 6nm. 6 n 2 , 6/mmm
3or3,sorG
c-axis 3 and/or 8 6 and/or 6
Four 3's. or 3's
Cube Fao--normals 2 and/or 2
Cube Body Diagonhls 3 and/or 3
(110)-axes:
4 ."A/",
".I-
i
e-axis 2 and/or
i
[Ilol
No1 axes-
=
2 end/& 9 -
i
-..-, -.s",,,"~
7
These axes are normal to the uniqne axis and pass through the midpoints of opposite edges of the unit cell; [ ] refers to a particular direct,ion, specified by the indices enclosed.
of s b , consistent with the point symmetry k d e r co&de'rat&n. Volume 44, Number 9, September 1967
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the unique axis is taken to be the crystallographic b axis (the so-called second setting; the less common first setting bas t,he c axis unique, aud so the symbol for t,he first setting would be Pll2/,1~). Thc symbd is usually shorn of redundancies to yield, for both settings, the short, or International, symbol, P2/m (11, 18). Taking another example, F4/11~3"/rn(International symbol, Fm:+m) refers to a face-centered cubic lattice (the cubic. symmetry being signaled by the 3 in the sea~iidaryposition), the point symmet,ry of the structure being t,hat uf a cube, m3m. The point symmetry elements referred to acubic unit d l and contailled in the full symbol are: Three 4-fold axes parallel to the eube face normals, (100) axes Three mirmr planes pil~.arllelto the cube faces, (10UI planeq where ( I refers tu a "form" of symmelry-dated planes Fonr :$-fold inversiort ares parallel tu the eube bwly diagwals, ( 1 1 1 ) ares Six Mold axes parallel todiagonnls through the midp
