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(Created on 28^{th} September 2005)

(Last revised on 18^{th} August 2009)

4.2 - POLYDOES WITH TWO RIDGES

4.3 - POLYDOES WITH TWO INTERSECTING RIDGES

What is a Polydo? It is a word coined by me.

What does it mean? As yet, nothing! That is why you will not find the word in a Dictionary, (not yet!). We have still to make it mean something. So let us start.

POLY-DO We'll start at the tail end. DO. DO is short for Donut. (Which again is short for Doughnut). No, we are not discussing donuts. Only the shape of a donut. Mathematicians call this shape a TORUS. But for you and me calling it 'donut' is good enough. A donut is circular in shape. Its cross-section is also circular. A Polydo is polygonal in shape. Poly means many. (or multi-). Or in our context, it means many-sided. Instead of a circular donut, if we have an octagonal (8 sided) donut, it would be called an OCTADO. The cross-section of an Octado is again an octagon. This is a drawing of an Octado.

So now, an Octado is an 8-sided Polydo. It has 8 straight segments and each segment meets the adjacent segment in a corner -. The corner of the Octado is itself a regular (equal-sided) octagon. When we refer to any of the vertices of the corner polygon we will call it a node -

Definitions:

1. Corner - the junction of two adjacent straight segments of a Polydo.

2. Node - any vertex of the corner polygon.

A 5-sided Polydo is a Pentado, a 6-Sided Polydo is a Hexado -. Etcetera, etcetera, etcetera. Get the picture?

Fig 01 shows an Octado

This Octado is coloured yellow. The colours have no significance in our discourse. They just help in a better visualisation of our figures. This yellow Octado will form the base - for a number of variants of Polydoes which we will be savouring later.

This is another drawing of a different Octado.

The difference between these two models is in the orientation of the cross-sectional octagon. Whereas the first model has none of its faces horizontal or vertical, the second model has 16 vertical faces, of which 8 are on the outer periphery, and the remaining 8 are on the inner periphery. Further, it also has 16 horizontal faces, of which 8 are on the top (all in one plane) and the remaining 8 are at the bottom (likewise in one plane).

It is coloured blue, which strictly means nothing for our current discourse. However, it may be mentioned here, that all the variants, discussed for the yellow Octado, could equally well be applied on the blue Octado. (with different colours of course, to avoid confusion)

In both these models, the straight segments between two adjacent corners form a series of octagonal prisms.

Now suppose we give a slight rotation to every alternate corner of the Octado, such that the nodes displace angularly to new positions which are, sort of, midway between the original node locations. Now, the straight segments will no longer be prismatic in shape. So we triangulate each segment as seen in this drawing.

The result is that now each straight segment is an anti-prism instead of a prism. We will designate this as an Anti-Prismatic Octado. (Or AP-Octado for short).

This terminology may be used for all anti-prismatic Polydoes Note that for prismatic Polydoes no prefix is necessary, but for anti-prismatic Polydoes, the prefix AP with a hyphen is used, as was done for AP-Octado. So, we can have AP-Hexado, or AP-Octado, or AP-Decado, - etcetera, etcetera, etcetera

It will be noticed that only even-sided Polydoes can be AP-Polydoes. This is due to the fact that every alternate corner of a Polydo is rotated, and the remaining alternate corners are left as they are, implying that there are an even number of corners. This is possible only for even-sided Polydoes.

Another class of Polydoes, where the prismatic and anti-prismatic segments alternate, would be Hybrid Polydoes. These are designated with a prefix H (for Hybrid). The next drawing shows an H-Octado.

Only Polydoes with (4n) number of segments (i.e. the number of segments must be divisible by 4) can have a hybrid form.

A five-sided Polydo is called a Pentado.

This is a drawing of one Pentado.

It is pentagonal in shape and the cross-section is also a pentagon. It has one set of nodes on the extreme outer periphery of the Pentado. The innermost faces are all vertical.

This is another drawing of a different Pentado.

It is different from the previous figure in that this figure has a set of nodes on the innermost periphery of the Pentado, and the outermost faces are all vertical.

The next drawing is that of yet another Pentado.

This one has a set of nodes on the uppermost part of the Pentado, and is flat-bottomed. Turn this one up-side-down and you have a flat-topped Pentado!

Interesting though Polydoes may be, they can be made much more fascinating. We can lend them an artistic beauty, which is almost sculptural. The possibilities are immense, and only some basic forms are presented here.

The corner of a simple Polydo is a regular (equal-sided) polygon. But it need not be regular (i.e. equilateral). Suppose we pull out one node of one corner polygon. A sort of a peak is created in the Polydo. Now suppose we do this to all the corner polygons, i.e. pull out one node each of all the corner polygons. And then join all the peaks so formed. A ridge is created. By carefully selecting which node of which corner to pull out, we can create fascinating figures. For instance, select any one node of any one corner to pull out. And at each successive corner the node selected is one up - (or one-down -). A beautiful ridge is created, which goes round the Polydo, weaving in and out. It is almost a-la-Mobius-strip (q.v. in any encyclopaedia).

This drawing shows a uniformly ridged Octado. The ridge is coloured blue (arbitrarily of course)

The next drawing shows a uniformly ridged Pentado.

The next drawing is one of a uniformly ridged AP-Octado.

Now this is amazing. It looks as if there are two ridges, but in fact, there is only one ridge going round the AP-Octado twice over, weaving in and out! This feature will be found on all AP-Polydoes. (This is sort of having a Mobius-strip which is slit along the middle, but still is one loop).

The above examples are those of a uniformly - ridged Polydoes, meaning thereby that the ridge is formed by creating all the peaks of the same height. This again need not be so. We may vary the heights of the peaks from maximum at one corner to zero at the opposite corner (or any other corner for that matter). And depending on which corner we choose to be of maximum peak height, we get a variety of beautiful figures. The next three drawings show the possibilities on an Octado. This drawing has a maximum peak on the outermost surface of an Octado,

The next one has a maximum peak on the top surface of the same Octado,

And this drawing has a maximum peak on the innermost surface of the Octado.

The next two drawings are for a varying ridged Pentado. This one has one maximum height and two zero height peaks.

And the next one has two maximum height and one zero height peaks.

(It may be noted that the base in this figure is a green pentado of Fig 6 with the ridge coloured red. Not that colours matter,)

Again, the maximum/zero height peaks may be located where you please, at whichever node of whichever corner you choose.

More interesting is a varying ridged AP-Octado. This drawing has a maximum height peak on the outermost surface.

The next drawing is also a varying ridged AP-Octado with a maximum peak at the uppermost surface.

And yet, another drawing of a varying ridged AP-Octado with a maximum peak on the innermost surface.

Let us now go back to the prismatic Octado. Suppose we were to create two ridges instead of one. We would now have a doubly-ridged Octado, both ridges being uniform. Note that this has two distinct ridges, unlike an AP-Octado, where a single ridge appears to be deceptively double! The two ridges are coloured with two different (arbitrarily chosen red and blue) colours.

The two ridges in the figure pass through two opposite nodes of each corner. This too, need not be so. But we will not go into that for the present. Let us be content with two equally spaced ridges - for the time being. We can even have 3 (or more as many more as we desire) ridges. Of course we may need higher ordered Polydoes - to accommodate more ridges say maybe a Dodecado or Icosado. Again, let us not go into those for now. Let's just stick to two ridges.

The two ridges may further be made of varying peak heights. Depending on where we choose to locate the maximum peak heights, we get quite a few variations.

The following figure has both the maximum peaks of the two ridges located on two opposite nodes at the same corner, the red one located on the outermost periphery and the blue one on the innermost periphery.

The next figure belongs to the same genre, but has the maximum peaks located lop-sidedly, off the outermost/innermost periphery

And the next figure has the maximum peaks on the top surface (blue) and the bottom surface (red)

The two maximum height peaks need not be located at different nodes of the same corner, but may be staggered. If the staggering is at full 180 degrees, the two maximum heights will lie on opposite corners. The stagger does not have to be 180 degrees, but any corner other than the same corner. The next three diagrams are for maximum peaks of the two ridges located at opposite corners.

This figure has both the maximum peaks located on the outermost periphery.

And the next figure has both maximum peaks located lop-sidedly towards the top surface.

And the next figure has both maximum peaks located on the top surface.

In the foregoing section, we had two separate and distinct ridges, necessarily parallel -to each other. In this section, we have two non-parallel ridges intersecting each other. It is easy to see that one ridge is a mirror image of the other. And that they intersect each other twice, at two opposite nodes of two opposite corners. Both the ridges in these cases are coloured the same colour. Of course the colour is unimportant.

This figure has the points of inte*rsection at the outermost and innermost
periphery.*

If the points of intersection are located lop-sidedly, one near the top surface and the other near the bottom surface, we get the following figure.

And for the intersection points located at the top and bottom surfaces we get the following figure.

A Pentado is a Pentado if it is a five-sided donut! (Say, that is a good quotable quote). A Pentado is just a Pentado, if its cross-section is also a five-sided pentagon. Imagine now, a Pentado having an eight-sided octagon as its cross-section. It is still a Pentado but now with a prefix. It is an Octa-Pentado. In other words, it belongs to a class of figures called Compound Polydoes.

Similarly, an Octado is an Octado - etcetera, etcetera, etcetera. Here is a drawing of a Penta-Octado.

The difference between an Octa-Pentado and a Penta-Octado will now be indelibly clear. See how simple it is. (I told you so).

Just for fun, here is a drawing of an AP-Penta-Hexado. (Who is afraid of prefixes!).

That's not all folks.