Such plots are named after Jean-Robert Argand —although they were first described by Norwegian—Danish land surveyor and mathematician Caspar Wessel — Stereographic projection Riemann sphere which maps all points on a sphere except one to all points on the complex plane It can be useful to think of the complex plane as if it occupied the surface of a sphere. Given a sphere of unit radius, place its center at the origin of the complex plane, oriented so that the equator on the sphere coincides with the unit circle in the plane, and the north pole is "above" the plane. We can establish a one-to-one correspondence between the points on the surface of the sphere minus the north pole and the points in the complex plane as follows.
Recently I came across a couple of studies that were among the first I carried out; some others have been lost and some are in private ownership. The studies which follow are all based on patterns I saw and explored, and I thought it might be useful to place them here for the record. Unfortunately there is no record of the examples they were taken from, so they will just have to stand as drafting exercises.
I should also admit that these sketches are based on the original photographs of the studies, most of the original drawings having been lost — the photographs being the only record I have of them, but which are in too poor condition to be reproduced.
The illustration above, from which the three following details have been taken, is of one of the original drawings I made in the early s. The width shown is around mm of the original. Although the paper has discoloured slightly with the passage of time, the draftsmanship can still be examined.
It is placed here to show something of the concentration that goes into study of this character of geometry as well as the accuracy I tried to maintain in drafting. Nowadays, drawing with computer programmes, there are still inaccuracies, but they are easier to deal with than when drawing with ink on paper.
At this degree of magnification it is possible to see how the grain of the paper, a smooth cartridge, interacts with the ink from the drafting pen. It is probable that the two pens used here were 0.
You can see there is some blotting at the beginning of lines — look at the red lines top right — as well as a little unevenness in line thickness. I believe I never went over any lines twice as that had a considerable effect on the outcome. In the central junction the red lines have created a star effect.
Note that it is not as symmetrical as I would have liked. This might have been due to the lines not drying before a new line was added.
Finally, this detail of one of the junctions illustrates one of the problems of joining lines at bends in the pattern. The pens have a circular nib and have to be held approximately vertically.
Where there is a change of direction of the line it is difficult to create a sharp corner. Considerable patience was needed in creating these drawings.
In analytic geometry, also known as coordinate geometry, we think about geometric objects on the coordinate plane. For example, we can see that opposite sides of a parallelogram are parallel by writing a linear equation for each side and seeing that the slopes are the same. Parallel lines remain the same distance apart over their entire length. No matter how far you extend them, they will never meet. The arrows To show that lines are parallel. The second way to form a plane is with a line and a point in that line. There are just two conditions. 1) the line must be perpendicular to the plane being formed (for an explanation of this concept, see Geometric Surfaces, Lines and Planes); 2) the point in the line must also be in the plane being formed.
This first exercise did not have any construction lines associated with it, as can be seen here. It was carried out to explore the relationship between stars based on ten-point geometry and there must have been construction lines before this was drawn, perhaps just pencilled in as these were all drawn manually rather than with a computer.
I have a vague memory that the origin of the pattern was on an Egyptian patterned door, but it is a fairly common pattern and might have come from a number of sources.
This next exercise was carried out in order to explore the construction of a pattern I had noticed as having been drawn with a single, continuous line. The setting out is easily established, being based on four point geometry, the diagonal lines are all base on joining intersections with themselves and with the horizontal and vertical centering lines.
This type of design is not uncommon, a variety of continuous lines being easily drawn and being suited to a design that interweaves. See the sketch three below. There are a number of patterns used to produce interlocking tiles, and this is just one of them.
Again the basis of the design is four-point geometry set out from the centering horizontal and vertical lines and then just repeated as far as is required. The underlying pattern does not have to be drawn in as much detail as I have shown, but there was something contemplative about this type of drawing that encouraged me to produce similar over-complex constructions, and that can be seen in many of the sketches on these pages.
This is the last of the sketch studies I recorded and was drawn, again, to establish the construction of the pattern on an Egyptian door.
Its head is on the left. As you can see, it is another design based on four-point geometry, but the difference here is that the lines that intersect are not as you might expect them to be, but often meet outside obvious junctions. Why this should be so is difficult to say, but it is only when you take the trouble to make investigations such as this that you are able to learn something of the workings of the minds of the original designers.
These two sketechs have been added as although they do not belong to the original drawings, I wanted to see how easily the sketch illustration, three above, could be worked up into — in the first instance — a relatively complex interweaving pattern.
I have added a drop shadow to improve the illusion of depth though this would not, of course, be found on a modelled example. Regrettably the sketch is not as symmetrical as it should be, but it does give an indication of how a basic framework may be readily developed into a solid design with the suggestion of a more complex character in its arrangement.
Although there are some advantages to working with a computer, it is unlikely I would have made the same mistake had it been drafted.Parallel lines remain the same distance apart over their entire length. No matter how far you extend them, they will never meet.
The arrows To show that lines are parallel. 1.
A line segment will always be the shortest distance between two points on a plane. The curved line and the broken line segments are further in distance between A and B.
2. If two points lie in a plane, the line containing the points lie in the plane When two planes intersect, their intersection is a line In analytic geometry, also known as coordinate geometry, we think about geometric objects on the coordinate plane.
For example, we can see that opposite sides of a parallelogram are parallel by by writing a linear equation for each side and seeing that the slopes are the same. staad pro caninariojana.com - Ebook download as PDF File .pdf), Text File .txt) or read book online.
MANUAL. The Icosahedron. The icosahedron has 12 vertices, 20 faces and 30 sides. It is one of the most interesting and useful of all polyhedra.
Buckminster Fuller based his designs of geodesic domes around the icosahedron. Geometry Here is a list of all of the skills students learn in Geometry!
These skills are organized into categories, and you can move your mouse over any skill name to preview the skill.