The current page refers many times to source files of Literka written for the program of Literka "Ruler and
Compass". It is not worthy to run these source files. Program is too large to use step-by-step option.
However, source files may be helpful for understanding the construction. To download
source files just click
download.
To download program "Ruler and Compass" see a link at the end of the current page. This program simulates constructions
with ruler and compass. That is why we can say that source files are a description of a construction with ruler and compass.
There are pages of Literka showing how to derive formulas for cos(k*pi/257). These are
Part I,
Part II,
Part III,
Part IV,
Part V,
Part VI,
Part VII.
Starting with a unit interval, which we assume has length 1,
we'll use all steps of these pages to construct an interval of length cos(2*k*pi/257) for k=84.
To avoid a confusion
something must be explained: We start with a unit interval, which has 100 pixels on computer's screen.
Then, for example, interval of length 2.5 corresponds to an interval of 250 pixels on a computer's screen.
This is independent of the resolution of a screen. The only difference is that a unit interval is smaller with higher
resolution. There is a new command "DrawDist" which draws values of distances.
It is a command of a new version of program "Ruler and Compass", which cannot be downloaded free yet.
If you have a free version of a program "Ruler and Compass", these commands must be erased for a program to
run.
Distances of a command "DrawDist" are measured in
pixels. If, for example, this command shows that an interval has length 567.66, the real length in our unit
system is 5.6766.
First we construct an interval of length t. The value t is described on the page
Part I.
To do this we need to construct roots of a quadratic equation from this page. We use the procedures of the
page of Literka
Construction of roots of quadratic equation
with ruler and compass.
These procedures are written for a program of "Ruler and Compass" in the included files "RootsPos.ltu"
and "RootsNeg.ltu" in a "zip" file for a download (link above).
In a source file, for a construction of t, we call a procedure "EqRootNeg(four,half)",
because the last coefficient of a quadratic equation is negative. Variables "four" and "half" are distances
of length 1/2 and 4 (On a screen they correspond to the intervals of lengths 50 and 400 pixels).
For example let us show intervals of lengths t and t1 (this is an output of a program "Ruler and Compass"):
Downloading link for source files is above.
Numbers are written with a command "DrawDist". For example, command for a second distance is
"DrawDist(60,83,$t#1#=%-,t1)".
It shows the number "-591.51". For the reasons mentioned above, the real value of t1 is -5.9151.
The same way we construct values t2,t3,t4,t5,t6,t7,t8,t9,t10,t11,t12,t13,t14.
Equations for these values are shown on pages
Part II,
Part III,
Part IV.
Using the command
"DrawDist" computer shows the following table:
To get real values, all these numbers must be divided by 100.
The next step is to construct values u1,u2,u3,u4,u5,u6,u7,u8,u9,u10,u11,u12,u13,u14,u15,u16. To to this
we write these numbers as linear combinations of just constructed numbers t,t1,t2...t14
using the matrix of the page
Part V
. For example
To construct linear combinations we use procedures of an included file "DistOp.ltu" of a program
"Ruler and Compass". Using the command
"DrawDist" computer shows the following table:
To get real values, all these numbers must be divided by 100.
We do not need all values xi and yi from the page
Part VI. We need only the following
distances, which we construct using equations from this page:
Using them and the equations from the page
Part VII
we construct the following distances:
The last distance is a value of absolute value of cos(2*84*pi/257), which we derive from the equation
(Now p12 is a negative number).
From this we receive that cos(2*84*pi/257) is about -0.4643...
Having this value constructed, it is easy to construct a regular polygon with 257 sides. We proceed the same
way as it was described on the pages
Construction of a regular pentagon and
Construction of a regular heptadecagon.
We draw two circles with the same center O, one with radius equal to our unit, second with radius
cos84=cos(2*84*pi/257). Draw a line L passing through O and take two points - point P0, which is an intersection
point of L with the larger cirle and point D, which is an intersection point of L with smaller circle.
We assume that P0 and D are on opposite sides with respect to O. Point P0 is our first vertex of a polygon.
Take a line perpendicular to L and an intersection point P84 with larger circle. This another point of our
polygon. Now we apply 255 (+2) times procedure, which is called "RegPoint" in source files. We find point
P168, which lies on a circle such that distance from P168 to P84 is the same as from P84 to P0. And so on and
so on.
Literka constructed only 7 vertices of this polygon. Received polygon looks like this:
If you have spare time, try to construct more vertices with source files of Literka.
Notice that we solved 24 quadratic equations for our construction.
Software that you really need.