Queries on Rational Numbers - 2
This post is being written with K Sheshadri and tries to consolidate things learnt following the last post...
We repeat the questions stated in the last post and summarize our findings on each...
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1. Can every rational number h be the hypotenuse of some right triangle of which the base and altitude are also rational? Or are there infinitely many such 'rational right triangles' with h as hypotenuse?
Answer: Yes. Indeed, there are infinitely many Pythagorean triples of integers {a,b,c} with c the hypotenuse. Obviously, there are infinitely many right triangles with unit hypotenuse {a/c, b/c, 1} all sides of which are obviously rational. Now right triangles with any other rational hypotenuse is got by scaling any of such unit hypotenuse rational right triangles by a suitable rational factor.
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2. a.Given any rational number P, are there infinitely many right triangles having perimeter P and with all 3 sides of rational length?
b. And for any specified rational A are there many right triangles having area A and with all 3 sides rational?
Part a has the answer "yes" and it follows trivially from answer to question 1 above.
As for part b, we have come to know about congruent numbers - a congruent number is a positive integer which is the area of a right triangle with rational length sides (https://en.wikipedia.org/wiki/Congruent_number). Numbers 1 to 4 are not congruent (remarkable, there is no right triangle with rational sides and area = 1). So, the short answer to part b is "not always".
Richard Guy's 'Unsolved Problems in Number Theory' asks: "How many primitive Pythagorean triangles have the same area?" and lists a few triples of such triangles and then asks "Are there any quadruples?"
Remark: Although there are infinitely many rational sided right triangles with hypotenuse 1, their areas do not exhaust the set of all rationals below 1/4 (1/4 is the largest area of general right triangles with hypotenuse 1).
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3. Among all the (uncountably infinitely many) *general* triangles all with some specified rational perimeter P and rational area A, are there infinitely many which have all three sides rational as well? And this same question can be asked with general quadrilaterals instead of general triangles.
Remark: We have no answer yet for the basic question 3. However for quadrilaterals and above, the answer is trivial: "Yes". Indeed, for each closed edge chain with n edges all of fixed rational length, we have a continuous range of area values possible - and this continuous range of areas contains lots of rationals. We can partition the desired rational perimeter into n rationals in infinitely many ways and for a large fraction of such partitions, we can form a closed and hinged edge chain for which the range of area values contains the desired rational area.
A further question - a 'freer' version of question 3:
4. Can every rational number be the area of *general* triangle(s) with rational length sides?
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Digression: Given a closed chain of rigid edges with hinges at vertices, how does one configure them on the plane to maximize the area within? IOW, if the length and order of the edges of a polygon are specified, how to determine its angles so that the area is maximum?
Eg: if the closed chain has 4 equal edges, the area is maximized by a square but the linkage can become rhombuses of any lesser area including 0. Obviously, for closed chains with 3 edges, the question doesn't stand.
One can also ask how to form a convex region of least area with such a closed linkage.
Answer: This question is nicely discussed at http://www.drking.org.uk/hexagons/misc/polymax.html
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Update (July 17th 2-19): We posted the above questions at MathOverflow and got this discussion. Thanks to those experts who shared their insights (https://mathoverflow.net/questions/335262/two-queries-on-triangles-the-sides-of-which-have-rational-lengths)
We repeat the questions stated in the last post and summarize our findings on each...
--------
1. Can every rational number h be the hypotenuse of some right triangle of which the base and altitude are also rational? Or are there infinitely many such 'rational right triangles' with h as hypotenuse?
Answer: Yes. Indeed, there are infinitely many Pythagorean triples of integers {a,b,c} with c the hypotenuse. Obviously, there are infinitely many right triangles with unit hypotenuse {a/c, b/c, 1} all sides of which are obviously rational. Now right triangles with any other rational hypotenuse is got by scaling any of such unit hypotenuse rational right triangles by a suitable rational factor.
--------
2. a.Given any rational number P, are there infinitely many right triangles having perimeter P and with all 3 sides of rational length?
b. And for any specified rational A are there many right triangles having area A and with all 3 sides rational?
Part a has the answer "yes" and it follows trivially from answer to question 1 above.
As for part b, we have come to know about congruent numbers - a congruent number is a positive integer which is the area of a right triangle with rational length sides (https://en.wikipedia.org/wiki/Congruent_number). Numbers 1 to 4 are not congruent (remarkable, there is no right triangle with rational sides and area = 1). So, the short answer to part b is "not always".
Richard Guy's 'Unsolved Problems in Number Theory' asks: "How many primitive Pythagorean triangles have the same area?" and lists a few triples of such triangles and then asks "Are there any quadruples?"
Remark: Although there are infinitely many rational sided right triangles with hypotenuse 1, their areas do not exhaust the set of all rationals below 1/4 (1/4 is the largest area of general right triangles with hypotenuse 1).
---------
3. Among all the (uncountably infinitely many) *general* triangles all with some specified rational perimeter P and rational area A, are there infinitely many which have all three sides rational as well? And this same question can be asked with general quadrilaterals instead of general triangles.
Remark: We have no answer yet for the basic question 3. However for quadrilaterals and above, the answer is trivial: "Yes". Indeed, for each closed edge chain with n edges all of fixed rational length, we have a continuous range of area values possible - and this continuous range of areas contains lots of rationals. We can partition the desired rational perimeter into n rationals in infinitely many ways and for a large fraction of such partitions, we can form a closed and hinged edge chain for which the range of area values contains the desired rational area.
A further question - a 'freer' version of question 3:
4. Can every rational number be the area of *general* triangle(s) with rational length sides?
---------
Digression: Given a closed chain of rigid edges with hinges at vertices, how does one configure them on the plane to maximize the area within? IOW, if the length and order of the edges of a polygon are specified, how to determine its angles so that the area is maximum?
Eg: if the closed chain has 4 equal edges, the area is maximized by a square but the linkage can become rhombuses of any lesser area including 0. Obviously, for closed chains with 3 edges, the question doesn't stand.
One can also ask how to form a convex region of least area with such a closed linkage.
Answer: This question is nicely discussed at http://www.drking.org.uk/hexagons/misc/polymax.html
------------------------------------------
Update (July 17th 2-19): We posted the above questions at MathOverflow and got this discussion. Thanks to those experts who shared their insights (https://mathoverflow.net/questions/335262/two-queries-on-triangles-the-sides-of-which-have-rational-lengths)
posted by R.Nandakumar @ 11:14 AM
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