The Diatomeans (Diatoms, Bacillariales) are a group of vegetable unicells of great diversity of shape. They also exist as linear or fan-shaped colonies (of the tectological order of an organ). They are notable for the possession of two silicified walls (frustules) inside the external cell membrane :   one of these frustules (epitheca) overlaps the other (hypotheca) like the lid of a box or tin. The cell looks very different according to whether it is viewed from above or below -- the valve-view -- or from the sides -- the girdle-view. Between the valve and girdle intercalary bands sometimes occur. In some species these bear septa projecting into the cell. The siliceous wall, especially in valve-view, is seen to bear extremely fine and complicated structures, often arranged in rows. These may consist of minute chambers, either open or closed above or below. If closed they are perforated by minute pores.
Frequently a groove or 'raphe' runs along the line of symmetry of the valve. The fine structure of this varies greatly in different genera. It is supposed that the extrusion of cytoplasm through the raphe causes the characteristic gliding motion which is peculiar to the Pennatae (A subgroup of the diatoms). [Strasburger's Textbook of Botany, New English Edition, 1976]

The next Figures schematize the global morphology of Diatoms.

Figure 1.   Every Diatom shell consists of two main parts, a 'box' (hypotheca) and its 'lid' (epitheca). These two parts generally are very much alike, they only differ a little in size.

In the literature on Diatoms the body axes of a diatomean cell have special names. We will present those given by HUSTEDT, 1956. We will use them to identify the corresponding promorphological axes. The next Figure depicts these axes. The cell is simplified such as consisting of two equal halves on top of each other. These halves represent the epitheca and hypotheca with their respective sides (considered to be identical) coinciding. The top part of the epitheca is the upper  v a l v e,  while the bottom part of the hypotheca is the bottom  v a l v e.  So the axis connecting top valve and bottom valve (the pervalvarous axis) is vertical in the Figure.

Figure 2.   Axes within the diatomean cell. The outline of this cell should here be imagined to be elliptical (not circular).
P-P' (running from top valve to bottom valve):   Pervalvarous Axis.
A-A' (representing the longer diameter of the cell) :   Apical Axis.
T-T' (representing the shorter diameter of the cell) :   Transapical Axis.

A diatom, when present singly (often they form loose colonies), is a single  c e l l  with a nucleus. The next Figure is a diagram of the inner parts of such a cell.

Figure 3.   Cell interior of Pinnularia.
K, Nucleus,   D, double bars,   Pb, Plasma bridge,   Pf, Plasma threads,   O, Oil drop,   Ch, Edges of the chromatophore plate, lying along the girdle side, but which is here omitted, in order to allow a view into the cell interior.
( After HUSTEDT, F., 1956, Kieselalgen (Diatomeen))

Figure 4.   The eight-sided amphitect bipyramid (Rhombic Octahedron). The equatorial plane is emphasized (yellow). The ideal cross axes (Euthini) are contained in the equatorial plane. The total symmetry content is :   three mirror planes perpendicular to each other, and three 2-fold rotation axes (coinciding with cross axes and main axis) each perpendicular to one of the just mentioned mirror planes. This symmetry content is symbolized by 2/m 2/m 2/m, where " 2 " stands for a 2-fold rotation axis, and " m " stands for a mirror plane.

In crystals the Rhombic Octahedron is the basic form of the Orthorhombic Crystal System :   The two radial cross axes together with the main axis form an orthorhombic crystallographic axial system, and this means that the three axes are different, but none of them stands out as a main axis.

So the Rhombic Octahedron is the point of departure when it comes to promorphologically assess a pennate diatom.
(The symmetry of) this Rhombic Octahedron, as the basic form of the Allostaura octopleura, we find materialized pretty accurate in Navicula and others. See the next Figure.

Figure 6.   Eunotia didyma.
The two valves of the shell. Valve form and position of raphes of the hypotheca (lower image) are the mirror image of the epitheca (upper image).
( After HUSTEDT, F., 1956, Kieselalgen (Diatomeen))

That valve that (at that moment of observation) lies  a b o v e,  i.e. towards the observer, is called A.

That valve that (at that moment of observation) lies  u n d e r,  i.e. away from the observer, is called B. See next Figure.

Figure 7.   Direction of observation of a diatom in valve view.
Turned-towards-viewer face is called A.
Turned-away-from-viewer face is called B.

During the consideration of the  w h o l e  cell, i.e. both valves (or both epitheca and hypotheca for that matter), applies :

Of A we always see only the  o u t e r  side.
Of B we always see only the  i n n e r  side.

We can never see the inner side of A (Recall that A is not a particular valve, A refers to the position with respect to the observer). This is so because when we flip the cell upside down (i.e. the set comprising its two valves), then A becomes B and vice versa.

When, on the other hand, we consider  o n e  s i n g l e  valve, then we can also always see the inner side of the valve after having it flipped upside down, i.e. we can see its outer and inner side. So we must discriminate between a consideration of one valve only and of the cell's two valves at the same time.

After having set all this we can now proceed with more detailed symmetry considerations.

Figure 8.   The two valves of Eunotia didyma (in valve view) as far as their symmetry is concerned. Each valve has as its only symmetry element a mirror plane perpendicular to the plane of the drawing (and as such it is symbolized by the hooked black line, i.e. the form of this line expresses the symmetry of the valve.
The appended short red line symbolizes the third spatial dimension, i.e. features of the valve situated along that dimension (the three spatial dimensions are perpendicular to each other). It also symbolizes the outer side or inner side of the valve. If the red line is directed upwards, as seen in the drawing, then we have to do with the outer side of the given valve, if, on the other hand, it is directed downwards, as seen in the drawing, then we have to do with the inner side of the given valve.
The red line being directed upwards as seen in the drawing, in fact is supposed to point right to the observer, while the red line being directed downwards as seen in the drawing, in fact is supposed to point right away from the observer. In both cases it is supposed to be  p e r p e n d i c u l a r  to the plane of the drawing.

Figure 9.   In order to be able to compare an outer side with an outer side (or an inner side with an inner side) we flip B about the axis indicated, which means that we rotate B about that axis by 180⁰. The outer side of it will now become visible Indicated by the red line being directed 'upwards'. So the result is :   the outer side of B.

Figure 10.   The outer side of A and the outer side of B are symmetric with respect to each other.
The red lines are supposed to stand perpendicular to the plane of the drawing (Recall that they represent the third spatial dimension). Here they both point right to the observer.

Figure 11.   The outer side of A and the outer side of B are symmetric with respect to each other.
The red lines are supposed to stand perpendicular to the plane of the drawing (Recall that they represent the third spatial dimension). Here they both point right to the observer.
In order to show this symmetry we create a mirror image of one of the valves (here B) and indeed see that the result is exactly identical (i.e. congruent ) to the other other valve (A). The mirror operation does not swap inner side with outer side (indicated by the red line).
When we do the same with valve A, then we will get a same result :
The mirror image of A will be completely identical (congruent) to B.

Figure 12.   Eunotia didyma.
The two valves of the shell. Valve form and position of raphes of the hypotheca (lower image) are the mirror image of the epitheca (upper image). There is yet another mirror plane as indicated by the red line (the plane is normal to the plane of the drawing, and normal to the other mirror plane.).

Figure 13.   Perspectivic view of a Rhombic pyramid.
Left image :   Rhombic pyramid with  t w o  antimers, representing the basic form of the  D i p h r a g m a.
Right image :   Rhombic pyramid with four antimers (basic form of the Tetraphragma).

Figure 14.   Pinnularia flamma.
The two valves of the shell. The course of the raphe is opposite on both valves. The valves as drawn must be imagined to be superimposed upon each other. The 2-fold rotation axis is indicated by a thin red line, and should be imagined to run between the valves. It is the only symmetry element of the cell.
( After HUSTEDT, F., 1956, Kieselalgen (Diatomeen))

Figure 15.   Symmetry diagram of Pinnularia flamma. Girdle-view. The two valves of the shell are symbolized by the two hooked lines, one above the other. The 2-fold rotation axis is indicated by a small blue circle in the center of the image, and should be imagined to run between the valves. It is the only symmetry element of the cell. The short red lines project into the third dimension (normal to the plane of the drawing). They represent features of the raphe, and at the same time show that the whole structure does not contain a center of symmetry (See next Figure).

Figure 16. Center of Symmetry.
The item possessing a center of symmetry (and only a center of symmetry) consists of two asymmetric faces related to each other by a center of symmetry, which means that every point of such a face is reflected in a same point, the center of symmetry (indicated by the red lines and their point of intersection. Every part of the item can also be found at the opposite side of that point at the same distance.

Figure 17.   Compressed symmetry diagram of Pinnularia flamma. Girdle-view. The 2-fold rotation axis is normal to the plane of the drawing, and is indicated by the small red circle.

Figure 18.   Diagram symbolizing the symmetry of Pinnularia flamma as depicted in Figure 14.

Figure 19.   Diagram symbolizing the flipping of valve B of Pinnularia flamma, about the axis indicated in blue, i.e. rotating it about that axis by 180⁰, resulting in the outer side of B.

Figure 20.   The outer side of A and the outer side of B.

If we find a symmetry operation that transduces the outer side of A and the inner side of B into each other, i.e. maps them onto each other, then we have found a symmetry element of the whole cell (here represented by its two valves). Let's try to transduce the inner side of B into the outer side of A :

Figure 21.   After rotating the inner side of B about the axis indicated in blue by 180⁰, we get the outer side of B (but with its apices exchanged). And now we see that this outer side of B (but with its apices exchanged) is exactly identical to the outer side of A. And this means that rotation of the inner side of B by 180⁰ about the axis indicated in blue, results in the outer side of A.

We will now subject the whole cell (represented by its two silicified valves) to a rotation by 180₀ about the axis as established above (i.e. as in Figure 20). The cell should be mapped onto itself. We'll start with a symmetry diagram of the the cell (its two valves) as was given earlier and which corresponds to Figure 14, and then subject this whole to the operation mentioned.

Figure 22.   Diagram symbolizing the symmetry of Pinnularia flamma as depicted in Figure 14. The 2-fold axis of rotation is indicated in blue. The whole structure is rotated 180⁰ about that axis. The valves are transduced as indicated by the arrows. Thereby B becomes A (green arrows), and A becomes B (pink arrows).

It is perhaps useful to dwell a little longer on the symmetry of Nitzschia sigma (Figure 25).
Also here we will symbolize the symmetry and shape of its two valves, like we did with respect to Pinnularia flamma :

Figure 27.   Symmetry scheme of Nitzschia sigma. Upper and lower valve (See Figure 25). The dashed line symbolizes the position of the raphe.

Figure 28.   Demonstration of the symmetry of a cell of Nitzschia sigma. Reflection in a mirror plane m followed by a rotation of 180⁰ about an axis perpendicular to m transduces the outer side of A into the inner side of B, which means that the whole cell has a center of symmetry. There are no other symmetry elements present.

Figure 29.   Triclinic Bipyramid, basic form of the Anaxonia centrostigma. Its only symmetry element is a center of symmetry, which is equivalent to a 1-fold rotoinversion axis, or to a combination of reflection and rotation as described above.

To  v e r i f y  the above assessment of the exact symmetry of the shell of Nitzschia sigma, namely that the only symmetry element it possesses is a center of symmetry, we apply this operation to the whole shell (cell). To do this we first reflect this whole shell as it is represented in Figure 27, and rotate the result 180⁰ about an axis perpendicular to the mirror plane. See next Figure.

Figure 29a.   When we apply the symmetry operation that corresponds to a center of symmetry to the whole shell of Nitzschia sigma as it is diagrammed in Figure 27, we will see that this whole shell will be mapped onto itself, i.e. the result is identical to that with which we began. So the shell as a whole has a center of symmetry.

The rotation by 180⁰ of the whole shell brings about the fact that the inner side of B, which, also after reflection, lies under, i.e. away from the beholder, will now come to lie UP and consequently becomes A, seen as the outer side of A. See next Figure.

Figure 29b.   Clarification of the rotation of the whole shell by 180⁰ as was done in Figure 29a.

Among the many Diatoms there are some that should be discussed briefly, because of their aberrant promorphology. These are Actinoptychus and Coscinodiscus. They are centric diatoms (Centrales).

Figure 32.   Photograph of the Diatom Coscinodiscus lacustris, valve view, made with the electron microscope of the Geological-Paleontological Institute of the University of Kiel (Germany). Original diameter 0.08 millimetre. This alga lives suspended near the surface of water with various salinity and is frequently found in the Baltic Sea. Photograph : Hans-J. Schrader.
( After JANTSCH, E., (1980) 1987, The Self-organizing Universe )