By Hand and Eye – need help
The first set of exercises in this book is to create rectangles of various dimensional relationships using only a compass and a straightedge. The basic set is 1:1, 1 ¼:1, 1/3: 1, 1 ½:1 , 1 2/3:1 , 1 ¾: 1 and 2:1
I had no problems creating these except for the 1 1/3:1 and 1 2/3: 1. Try as I may, I could not accomplish this assignment. I gave this problem along with a compass and straightedge to several of my very bright young guests at Thanksgiving. None could create the 1 1/3:1 or 1 2/3:1 by using multiple circles as directed by the text.
To add to my frustration, the book assumes you will do this easily and then states these are the “basic building blocks needed to construct forms in your head and at the workbench”.
Any help in providing the steps to create the 1 1/3:1 or 1 2/3:1 rectangle by using multiple circles will be greatly appreciated and allow me to proceed.
Replies
what is the book title?
I personally know from many (fruitless) hours trying to divide an angle into thirds that it cannot be done - later confrimed by an instructor (called a teacher back then).
So I need to read this 'book' saying that it can be done with a compass and a straightedge (I have both).
Forrest
By Hand & eye
It is all
By Hand & eye
It is all about not using dimensions for design, but rather using ratios as was done for thousands of years before industrialization moved the design process to dimensional relationships to accomodate mass production by machines.
Why?
Not to be contrary, but what is the point of dividing an angle by thirds with a dividers? Not that it isn't a neat mental excersize, and might be handy once in a while, but it hardly seems that "these are the basic building blocks needed to construct forms in your head and at the workbench”.
Craftsmen of way back when used to layout lots (everything?) of things geometrically, with a unit measurement, straight edges, strings, dividers, and trammels. The advantage was that you could instantly scale thing up and down by changing that inital unit of measurment. Things like furniture, harpsichords, pianos, paintings, and buildings got designed and laid out this way. Their construction tended to be around the golden mean, the square roots of 2 and 5, and Fibonacci series though, as they are easy to do geometrically.
OK, now to be useful, a quick web search shows that trisecting an angle is theoretically impossible with just a dividers and straight edge. Apparently this is a big raging debate in math nerd circles and has been for 2000 years. The simple solution (and the one that I would use) would be to used a graduated straight edge (i.e. a ruler) equidistant from the vertex and divide it out using the graduations.
As noted in my reply to my post. I was not trying to divide an angle but rather create a rectangle with the ratio of 1: 1 1/3 and
1:1 2/3
I have found the solution and posted a PDF showing the process.
My last math class was over 50 years ago, so I am not a math wiz. However the solution I found to my problem could easily divide an angle into thirds as well. Using a compass, mark the exact same length off on the two legs. Draw a straight line between the two points and use the process in the PDF to divide the straight line into however many equal parts you like.
Not there yet
Sorry to b long but am on trip - no computer. The PDF method ONLY works if the small segments on the upper and lower line are exactly equal. Why? Well, ALL 3 LINES ARE EQUAL LENGTH. The upper and lower lines are nothing but radii of a circle whose radius is the original line. U can divide a line into thirds - but not an angle with only a compass and straightedge.
Not there yet
Sorry to b long but am on trip - no computer. The PDF method ONLY works if the small segments on the upper and lower line are exactly equal. Why? Well, ALL 3 LINES ARE EQUAL LENGTH. The upper and lower lines are nothing but radii of a circle whose radius is the original line. U can divide a line into thirds - but not an angle with only a compass and straightedge.
To be clear
Your PDF "solution" cannot work to divide any given line into any number of equal segments for the reason I have given.
But you stated that the same "solution" could be used the divide any angle into thirds. There is no proof of that, true, but as stated above 2000 years of trying have not shown a solution.
If you are interested in actually dividing any line into any number of equal segments send me a PM with your email and I will send it to you. But, please, do not use the PDF version in any design.
Forrest
The problem had nothing to do with dividing an angle. I need to be able to make a rectangle equal to 1 1/3 and 1 2/3 D : 1
I needed to be able to find 1 1/3 D and 1 2/3 D using only a compas and ruler. I have learned that it can be done and I have moved on. attached is the process
Still having a problem with this?
Just worked thru the example you posted. Worked fine for me. What were you trying to do, divide that line into 3 sections or 5 sections instead of an even number? The method should still work. All the method does is generate a parrallelagram with your starting line as a diagonal. Then it generates identical triangles above and below that diagonal.
oh, well
In the "pdf solution" if you cannot see that
line A-D is the same length as line A-B and the same length as line C-B
which REQUIRES
that line A-1 equals exactly 1/4 of line A-D
and line 1-2 equals exactly 1/4 of line A-D
and line 2-3 equals exactly 1/4 of line A-D
and line 3-D equals exactly 1/4 of line A-D
for the segments on A-B to marked into 4 equal lengths
you might consider further education in geometry and trigonometry.
Again, please do not use this 'method' to sub-divide a line into equal lengths.
Check out http://www.youtube.com/watch?v=E65w7M8VSAQ to learn how.
Forrest
Yes..........but
line A-D is the same length as line A-B and the same length as line C-B......agreed............
But,
in the example, the line segments, A - 1; both 1 - 2's; both 2 - 3's; & 3 - B are all 15/16". Line segments 3 - D & C - 1 are each 19/32. The total length of the line A - B is 3 15/32". The segments of line A - B from the construction are 57/64. That's 1/4 of the total length.
On the example I did for myself, the length of line A - B was 6", my line segments A - 1; both 1 - 2's; both 2 - 3's; & 3 - B were all a tad bit less than 1 1/4", leaving line segments 3 - D & C - 1 as roughly 2 11/16. That breakup is considerably different from having each at 1/4 of the total length of line A - B. The result of the construction was to divide line A - B into 4 equal lengths of precisely 1.5", reading at least to 32nds.
As they say in the texts, "The analytical derivation is left as an exercise for the reader."
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