h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

The prefered way to draw a pentagram is to draw a circle with centerlines at 90 degrees to each other crossing in the center. looking at an analog clock one line goes from 12 to 6 and the other from 9 to 3. The point were the two line cross we will call center.

First divide the line from center to 3 in half, this can be easily done with dividers set at over half the distance and drawing an arc from both the center and 3 points. A line between were the arcs cross above and below the line will divide the center - 3 in half.

Next draw an arc with the center at the half way point from center - 3 and the radius set at 12 draw this arc through the 9 - center line.

Almost there

Now draw one more arc with the center at 12 and the radius set by the point were the last arc intersects the 9 - center line and draw this arc through the circle between 9 and 12.

This radius is 1/5 of the circle and can be drawn from the above intersection as center to find the next point. Repeat and you will have your pentagam without the math!

Just another way to solve your problem, I have used this method many times. If you want to use it please feel free

F S

Or you could just use a compass, a pencil, and 5 72 degree angles.

AK That would work great if you have a 72 deg angle handy. Then again I can load up my drafting program, select draw polygon and type 5 for the number of sides.

This is intended for someone who only has a few standard drawing tools at hand.

F S

This method is very nice, and does have much more geometrical "beauty"- I thought there might be a method of that kind, but I was'n in the mood for that kind of geometry, but I still have some reservations to it:
First, I'm not sure it's prefered, but that's not important.
Second, I tried it twice, and didn't quite get a pefect regular pentagram, that is when I looked at the pentagon inside it didn't look the same from every direction. This could have a few reasons-
1. The method is wrong.
2. I was'nt accurate enough.
I personally think that the secon option is right, but it is derived from a problem at the method. (I think that)This method uses too many guiding points which rely on each other, therfor in each step the mistake grows bigger. My method has the same problem, but for some reason it was less noticeable. sorry if you feel I'm babling, because I get this feeling.

Anyway, would you like me to add your method to the entry?
Awaaiting your reply, Ur.

P.S. Do you have a geometrical demonstration for it? If not I'll try to find out myself.

The method I explained came from an old drafting textbook, reworded into plain english both to avoid just copying someone else's work and to avoid the diagrams required for the explaination in the source.

I believe the irregularity in your attempts is probably a small error in each step which can accumulate ( Murphy's Law ).

If you have a good drawing program on your computer you might get better results. The 'prefered' line came from the source and I agree it should probably be dropped.

One way to check the layout is after marking each point on the circle strike another arc between the last two points and see if the marks match, this will quickly show if the final arc is too large or too small before completing the pentagram.

From reading your entry I got the feeling that you wanted to show others how to draw a pentagram with limited resources, wich is why I gave the other method, you are quite welcome to include it as another method and the protractor method might add another alternative for those who have a protractor available.

I think your entry would be most useful with a few different methods of drawing the penagram.

BTW I think I typed dividers when I meant compass, so if you include this please correct.

F S

Theres some relationship between the golden ratio and a pentagon. you draw a line from a point on the pentagon to the one two over from it. do this twice so that the two lines intersect. I *think* that where one of the lines is cut in two, the ratio between one of the sides and the other is the golden ration(1.618033)
The golden ration also shows up a lot in pentagrams other stuff...
I'm not sure if that helps you create on though...
There are several origami ways of creating one, including one that uses a hexagonal pyramid's shadow!

Oi theres already stuff abotu the golden ration on that page, I didn't remember!

Well, I didn't know that the number I called is also called "the golden ratio"...
Can you tell me more about this origami thing (or direct me to a site that deals with that?

For FloridaSailor- 1. from what I understand "dividers" and "compasses" are the same thing (according to Word's Hebrew-English dictionary) and are also called "calipers".
2. About you're second paragraph- that is what i meant in my reply earlier. You just said it better than I. I'm not so sure about it's connection to Murphy, but never mind.
3. What book is that?

Ok, I have about 3 books with stuff about this...

First of all the golden ration has lots of cool properties, including, I think: its used to make seashell spirals, it has a relationship to fibonacci patterns, a relationship to rectangles(golen rectangles), and other stuff
its what you get when you take the line from the center of one side of a square to a corner, and rotate it on the axis on the side of the square down, creating a rlonger rectangle. Thats a golden rectangle.

If you divide every set of two consecutive numbers in a fibonacci pattern (1 1 2 3 5 8 13 21 34 55 89 144 etc...) by the next one up, the results will get closer and closer to 1.618033 as the numbers get higher.
The golden ration = 1+(1/(1+(1/(1+1/(1+1/(1+1/(1... which goes on forever

ITs the relationship between the long and short lines in a regular pentagram. I sure you can find more on google...
The origami thign though is so very hard to explaing you'd need the diagrams to do it, but thats probably on the web somewhere too...

1. from what I understand "dividers" and "compasses" are the same thing (according to Word's Hebrew-English dictionary) and are also called "calipers".

In my own learing a compass has a pen or pencil point on one leg and a metal pin on the other, whereas dividers and calipers have metal pins on both legs and are used for measuring rather than drawing. This may be US specific, I'm not sure.

2. About you're second paragraph- that is what i meant in my reply earlier. You just said it better than I. I'm not so sure about it's connection to Murphy, but never mind.

Murphy's law is an old engineer's joke, 'anything that can go wrong will go wrong at the most inopportune time' such as at the demonstration.

3. What book is that?

The specific book I used is "Technical Drawing" fifth edition by Giesecke, Mitchell, Spencer and Hill copyrite 1967
it is an old textbook from technical school.
As I said earlier I only used the method not the actual description and this is standard geomotry so it is public domain ( the use of the points as hours on a clock are totally my own etc. )

If you really want to play with numbers try trigging out the various triangles formed.


F S

http://dmawww.epfl.ch/roso.mosaic/dm/murphy.html for many many Murphy's laws.

Great link AK!

F S

Interesting.
"ITs the relationship between the long and short lines in a regular pentagram" -this I realised myself, that is why I wrote this entry. This golde ratio is basiccaly the positive solution the the equation X^2=X+1. About the Murphey law- I know what it is, I'm just not exactly sure it connects. The thing that does deal with such irregularities is the theory of chaos, as far as I know- for example In the pentegram I drawed with my method I took the pentagon that is created and used it to draw anothe pentagram inside of the first and did it a few times until it became impossible. By the third one irregularities started to shoe and the last one was pretty croocked. This is aform of chaos, I think.

>About the Murphey law- I know what it is, I'm just not exactly sure it connects. The thing that does deal with such irregularities is the theory of chaos, as far as I know- for example In the pentegram I drawed with my method I took the pentagon that is created and used it to draw anothe pentagram inside of the first and did it a few times until it became impossible. By the third one irregularities started to shoe and the last one was pretty croocked. This is aform of chaos, I think.<

The reason I mentioned Murphy's Law is it would seem logical that a number of small errors will cancel each other out however often they all add in the same direction.

If for instance you need to layout 10 spaces a unit of 5 apart ( I am avoiding naming the unit to avoid US - Metric confusion ) and there is an allowed error of .01 the over all distance from one end to the other should be 50, however it can be from 49.95 to 50.05. If this error is too large for the part to work it must be laid out differently, usually measuring each point from one end. The smaller the part the smaller the tolerance, a house has a much larger tolerance than a watch.

Nothing in the real world can be an exact distance, no matter how close you get a more accurate measuring device can be developed to show the error even if it is almost molecular. In industry this error is called &#8216;tolerance' and is usually shown on drawings or it may be a company policy. As a rule the smaller the tolerance the more expensive the piece is to produce.

This is also part of the origins of chaos which is more from a mathematical point of view while I tend to look at it from a physical layout viewpoint. When your standing on a shop floor with 3 people standing around waiting for you to fix their layout theory doesn't count, you have to make it work.

F S

I'm not quite sure but I think the golden ratio is expressed with the Greek letter Phi... but i'm not sure about any of this, so...

ah ha! A643844

I think I've read about a nice method used by Albrecht Dürer - can't remember how it went just now, but it was quite elegant.

Only, according to my source, you don't *quite* get a regular pentagram, though the error is really tiny indeed (probably less than the errors I would commit actually drawing the picture to be honest ).

I'll try and find it for you if you wish.

Here's one origami method to create a pentagon fro a square, thoguh not the one I use: http://www.ganymeta.org/~darren/origami/shape.pentagon.gif

after that all you have to do to create a pentagon is cut out a few lines.

The simplest way to make a pentagon by folding paper that I have found is what I call the "Bus Ticket" method. You require a rectangle of paper that is about seven times longer (like a long bus ticket) than it is wide. Simply make a loose overhand knot with the strip and gently pull the two ends, until the pentagon shape appears. Gently flatten and adjust until the ends and corners form a pentagon with the two ends forming a wide "V", and the pentagon shape below. Fold the two ends of the "V" along the top of the pentagon. Hold up to the light, and within the pentagon there is a pentagram, and a little pentagon inside that.