Constructibility of a Perfect Venn Diagram

In summary, the conversation discusses the possibility of constructing a Venn Diagram or a pair of overlapping circles where the area of the lune of each circle is equal to the other and to the overlapping region. The speaker has worked out the mathematics involved and believes that the required angle θ may not be constructible. They suggest constructing a base-line and bisecting it to solve the problem, but acknowledge that this may not be possible due to the limitations of dividing lengths.
  • #1
CarsonAdams
15
0
Is it possible using only a straight edge and a compass to construct a Venn Diagram or a pair of overlapping circles in which the area of the lune of each circle is equal to the other and to the area of the overlapping region?

(I've worked out most of the mathematics from calculating the area of the sector (cad) and subtracting the area of the triangle (cad) to achieve the segment of the circle created by the chord (cd) which accounts for half the total overlap. Then to achieve a useful equation, I calculated that, obviously, the area of the circle divided in half must equal twice the area of the single segment created by the chord.)

After simplifying, it would seem that to construct the required angle θ the equation
θ-sin(θ)=π/2
would have to be solved (in radians), which would seem to denote that it is not a constructible angle.

But questions of constructibility usually require someone much more clever than I, so am I missing a simplifying way to achieve this? and is it fair to say that the only way to solve the above equation is through the use of numerical methods?

*if it would be useful to review the full calculation, I can post that as well- it's just a bit tedious
 

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  • #2
To construct the diagram so all three areas are equal - you need to construct a base-line, then bisect it - the circles, of equal fixed radius (1 unit), are to intersect on that line so their centers are on the base-line. So the problem is to relate the location of the centers of the circles to the areas.

The area of the intersection region is twice the area of the circular segment cut off by the bisecting line.
Using the diagram in wikipedia: the distance from the bisector to each center is ##d##.

... I don't know if this can be constructed... but it would amount to how finely you could divide lengths up.
 

1. What is a Perfect Venn Diagram?

A Perfect Venn Diagram is a mathematical representation that uses overlapping circles to show the relationship between different sets of data.

2. How is the Constructibility of a Perfect Venn Diagram determined?

The Constructibility of a Perfect Venn Diagram is determined by the ability to accurately and clearly represent the relationship between the different sets of data using circles of equal size and accurate placement.

3. What are the requirements for a Venn Diagram to be considered "perfect"?

To be considered perfect, a Venn Diagram must have circles that are equal in size, accurately placed, and the overlapping areas must accurately represent the relationship between the data sets.

4. What are the limitations of constructing a Perfect Venn Diagram?

There are a few limitations to constructing a Perfect Venn Diagram, including the fact that it can only be used to represent two or three sets of data, and it may not be the most efficient way to represent complex relationships between large sets of data. Additionally, the accuracy of the diagram may be affected by the precision of the tools used to construct it.

5. How is the Constructibility of a Perfect Venn Diagram relevant in scientific research?

In scientific research, the Constructibility of a Perfect Venn Diagram is important because it allows researchers to visually represent the relationships between different sets of data in a clear and concise manner. This can aid in analyzing and understanding complex data sets, and can also help in identifying patterns and connections between different variables.

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