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This is in continuation of my earlier post here.

On pg.#6-7, question #5 is about uniform growth used to form one rectangle from another by shrinking or expanding $x,y$ coordinates. The different parts of the question are attempted below with need for inputs in part (b), & vetting for the rest:
a) Describe all the points in the lh-plane that represent the rectangles uniformly grown from the rectangle $(3, 2)$.
-> $(3\pm i, 2\pm i), i\in \mathbb{R}$
b) Given the rectangle $(3, 2)$ and the rectangle $(5, 3)$, can you find a rectangle that can be uniformly grown from each of them? Explain.
-> Incomplete attempt - Need take differences in $x,y$ coordinates of the two points, and form a line joining the two, with slope =
c) If one rectangle is uniformly grown from another, can they be similar? Describe all such pairs of rectangles.
-> Yes, the answer is $(a\pm i, b\pm i), i\in \mathbb{R}$, with $(a,b)$ the start point of the rectangle.
d) Is it possible to grow (or shrink) uniformly a square from any rectangle? Explain.
-> No, except rectangle not possible to generate for 'possibly' one instance where the two points become equal. Otherwise, if the start rectangle is a square; then always possible.
Although, am still to prove that from a rectangle there can be at max. one instance where square can be produced. Request idea for that apart from part (b).

jiten
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Guide:

  • If they grow at the same rate, then they can be written as $(3+t, 2+t)$ for some $t\ge0$. The same parameters are used in both entry of the coordinate so that the growth is uniform.

  • To answer $(b)$, try to solve whether you can find a solution for $(3+s, 2+s)=(5+t,3+t)$.

  • For part $(c)$, the question is equivalent to consider rectangle being characterized by $(a,b)$ and the growth rectangle is $(a+t, b+t)$, is it possible that they are similar? if so, what are the characterization of $a$ and $b$.

  • For part $(d)$, You might like to fix a particular rectangle and show that you can't grow a square.

Siong Thye Goh
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  • Thanks. For each part, the attempt is: (a) similarly for shrinkage, $t\lt 0$; (b) get both $s-t=2, s-t=1$, i.e. a pair of simultaneous linear equations with parallel lines, so no solution; (c) such similarity demands the constraint : $\frac{a+t}{b+t} = \frac{a}{b}\implies b(a+t) = a(b+t)\implies t(b-a)=0\implies t = 0$, so no further pair; (d) taking $(3,5)$ as rectangle, and adding/subtracting uniformly (i.e., to both co-ordinates) any real value $i$, it is seen that the proportions never change. But, (c) & (d) both are not possible, seemingly they are opposites! – jiten May 18 '18 at 06:31
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    For part $(c)$, if you get $t(b-a)=0$, there is another possibility besides $t=0$. no idea what is going on for your working in part $(d)$. – Siong Thye Goh May 18 '18 at 06:43
  • Yes, either $t=0$, or $a=b$. For part (d), need prove that (in contrary to part (c)) that the original ratios change, i.e. given point $(3,5)$ or $(a,b), a\ne b$ in general; to get a square from it tantamount to $(a+t)= (b+t)\implies $ the only possibility of $a=b$, i.e. only a square can lead to a square. – jiten May 18 '18 at 07:02
  • Please vet my attempt in Q.6(b) on page #7 in first chapter (https://www.worldscientific.com/doi/suppl/10.1142/7810/suppl_file/7810_chap01.pdf). Its first part asks for getting a point with coordinates $(4,5)$ by expanding uniformly, from a line with aspect ratio $(=\frac{height}{width})=0.5$, or inclination of line ($=\arctan 0.5$). My attempt: $\frac{1+t}{2+t} = \frac45\implies 5+5t=8 +4t\implies t =3$. Second part asks if a square can be drawn from this family. My attempt: $\frac{1+t}{2+t} = 1\implies 1=0\implies$ : It is not possible. – jiten May 18 '18 at 11:52
  • Kindly see my related posts : (a) for 'Proportional growth' at : https://math.stackexchange.com/q/2786300/424260. I am totally confused and lacking for ideas there in parts (c), (d) that asks for alternating the uniform and proportional models. ----- (b) for 'distance function' at : https://math.stackexchange.com/q/2786383/424260, that seeks to find a quantitative measure for closeness among rectangles. – jiten May 18 '18 at 14:17
  • I request some guidance on my previous comment for two posts, that are still without response / comment on mse. – jiten May 18 '18 at 19:36
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    i will look at them when i'm freer. – Siong Thye Goh May 18 '18 at 20:06
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    remark: i will be travelling... let's see when do i have time. time scale expectation: could be days. – Siong Thye Goh May 19 '18 at 00:46