Does neusis constructions include cardioid?

[DaTario]

Hi All,

I would like to know if it is correct to say that neusis constructions (which allow the use of conchoid of Nicomedes, for instance) allow the use of cardioid, which is also a conchoid.

Best wishes,

DaTario

 

[jedishrfu]

I don't think you can do it. The Wikipedia article shows construction of the cardioid by rolling a circle around a circle.

https://en.m.wikipedia.org/wiki/Cardioid

Later on in the same article there was another scheme via tangent line and a sliding point but no neusis constructions were shown.

 

[DaTario]

But consider the figure below, from a paper called The Cardioid (The Mathematics Teacher 52(1) 1959 pp. 10-14) showing a construction very similar to that of a conchoid of Nicomedes.

 

[jedishrfu]

Perhaps one of other mentors can comment here like @fresh_42 or @Mark44

So you want to know if the image construction method is a neusis construction?

 

[Mark44]

Perhaps one of other mentors can comment here like @fresh_42 or @Mark44

No clue.

 

[jedishrfu]

The way you'd have to approach answering your question is to determine what constitutes a neusis construction. As I understood it, you needed a fixed point for your marked ruler to rotate around, a starting curve or line ti indicate the midpoint and then an endpoint on the ruler to indicate the trace of the new curve.

If you take cardioid and can construct it that way with the circle being the starting curve then we'd have to say yes the cardioid is constructable by neusis methods. The example you showed doesn't seem to be using the marked ruler and seems to have an extra joint in there and that's why I think its not constructable by neusis methods.

 

[jedishrfu]

But consider the figure below, from a paper called The Cardioid (The Mathematics Teacher 52(1) 1959 pp. 10-14) showing a construction very similar to that of a conchoid of Nicomedes.

View attachment 114705

A key question is whether PO and OM are of equal length as you trace out the cardioid? Does it describe the construction method in the text that you didn't post?

Also I need to say that this subject is somewhat interesting. I had never heard of the neusis construction method before. Thanks for sharing.

It might make an interesting PF Insight article here. The history on wikipedia mentioned that the Greeks used this construction method first and then abandoned it in favor of straight edge and compass perhaps because of the later's simplicity (unmarked ruler) and rigidity (no ruler rotations about a point) whereas the neusis was relegated to problems not amenable by straight edge and compass constructions.

I got interested in Origami and discovered that origami methods can do the cube doubling and angle trisection problems which really amazed me.

 

[DaTario]

Yes, PO = OM. It is the fundamental property of conchoid which is present in the cardioid. I confess that the above linkage (name given to this general kind of mechanical device) does not communicate the geometry very clearly. One important fact is that in the point O is marked a kind of small straw (or tube) which is fixed in O and is allowed to rotate. The rod P-O-M goes inside this "tube".

As far as I am aware of, neusis constructions appear as a way to solve problems like the trissection of an angle. I am trying to recover a nice reference in which there is a good account of these methods of the Greeks. They were ranked according to how challengeable they are. The use of just straight edge and compass was considered the method which demanded the highest intelectual level. But it is a fact that this method has limitations.

 

[jedishrfu]

How about this as a starting point?

https://en.wikipedia.org/wiki/Neusis_construction

and this pdf with further references to check out:

http://math.ucr.edu/~res/math153/history09c.pdf

 

[DaTario]

Very nice references. The Wikipedia I was already aware of. But the other one is new to me. I may say now that I am convinced that the use of a cardioid represents a neusis construction. I will follow the sites your second reference recomends to see if one of them uses cardioid. But anyway, thank you very much.