# #2^N# unit circles are conjoined such that each circle passes through the center of the opposite circle. How do you find the common area? and the limit of this area, as #N to oo?#

##### 3 Answers

, N = 1, 2, 3, 4, .. Proof follows.

#### Explanation:

Before reading this, please see the solution for the case N =1

(https://socratic.org/questions/either-of-two-unit-circles-passes-through-the-center-of-the-other-how-do-you-pro).

For N = 1, the common area

the equal circular arcs of the two circles, each subtending

at the respective center. The center O of this oval-like area is

midway between the vertices

subtended by each arc at O, on the common chord, is

See the 1st graph.

The common area for N = 1 is

When N = 2, There are

boundary of the common area, with the same center O and

vertices

is

nearly.

The common area for N = 2 is

Note that the first term is angle in rad unit.

The general formula is

Common area

, N = 1, 2, 3, 4, ..

2-circles graph ( N = 1 ):

graph{((x+1/2)^2+y^2-1)((x-1/2)^2+y^2-1)=0[-2 2 -1.1 1.1]}

4-circles graph ( N = 2 ):

graph{((x+0.354)^2+(y+0.354)^2-1)((x-0.354)^2+(y+0.354)^2-1)((x+0.354)^2+(y-0.354)^2-1)((x-0.354)^2+(y-0.354)^2-1)=0[-4 4 -2.1 2.1]}

The common area is obvious and is shown separately ( not on

uniform scale). Here, y-unit / x-unit = 1/2.

graph{((x+0.354)^2+(y+0.354)^2-1)((x-0.354)^2+(y+0.354)^2-1)((x+0.354)^2+(y-0.354)^2-1)((x-0.354)^2+(y-0.354)^2-1)=0[-0.6 0.6 -0.6 0.6]}

(to be continued, in a second answer)

Continuation , for the second part.

#### Explanation:

The whole area is bounded by

common center O. Each

subtends an

=

at the common center O.

So,

Let this arc subtend an angle

circle. The graph shows the common center O, the arc AMB and

the radii CA and CB, where C is the center of the circle of the arc.

The common area

CAB + area of the

on the left at (-0.5, 0). M is ( 0.5, 0), on the middle radius.

graph{(0.2(x+0.5)^2-y^2)(x^2-y^2)((x+0.5)^2+y^2-1)=0 [0 1 -.5 .5]}

Let

area of sector CAMB =

area of

and area of #triangle OAB = 1/2( base)(height)

As AB is the common base of

As

limit of the common area is

This is the area of a circle of radius 1/2 unit. See graph.

graph{x^2 + y^2 -1/4 =0[-1 1 -0.5 0.5]}

For extension to spheres, for common volume, see

https://socratic.org/questions/2-n-unit-spheres-are-conjoined-such-that-each-passes-through-the-center-of-the-o#630027

Continuation, for the 3rd part of this problem. I desire that this for circles, extended 3-D case for spheres and all similar designs are classified under "Idiosyncratic Architectural Geometry".

#### Explanation:

Continuation:

If the condition is that each in a triad of unit circles passes through

the centers of the other two, in a triangular formation, the common

area is

central common area.

graph{((x+0.5)^2+y^2-1)( (x-0.5)^2+y^2-1)(x^2+(y-0.866)^2-1)=0[-4 4 -1.5 2.5]}

This can be extended to a triad of spheres, and likewise, a

tetrahedral formation of four unit spheres. Here, each passes

through the center of the other three, and so on.

Indeed, a mon avis, all these ought to be included in

Idiosyncratic Architecture.