@Markiyan Hirnyk 

A projection of a triangle is a triangle. As can be seen from the first plot (with code) this is false. So, MMA is totally wrong.

@Markiyan Hirnyk 

The MMA plot does not seem to be correct.
Can you plot the (x1,x2) projection?

The 13 regions obtained with Maple are:

Region1
{442/7 < x2, x1 < 80, x2 < 64, 381/2-(7/4)*x2 < x1}
Region2
{432/7 < x2, x1 < 78, x2 < 62, -(7/4)*x2+186 < x1}
Region3
{414/7 < x2, x1 < 78, x2 < 60, -(7/4)*x2+363/2 < x1}
Region4
{438/7 < x2, x1 < 75, x2 < 442/7, -(7/4)*x2+369/2 < x1}, {442/7 <= x2, x2 <= 444/7, 74 < x1, x1 < 75}, {x1 = 74, 442/7 < x2, x2 < 64}, {74 < x1, 444/7 < x2, x1 < -(7/4)*x2+186, x2 < 64}, {x1 = -(7/4)*x2+186, 444/7 < x2, x2 < 64}
Region5
{426/7 < x2, x1 < 75, x2 < 430/7, -(7/4)*x2+363/2 < x1}, {430/7 <= x2, 74 < x1, x1 < 75, x2 < 62}, {x1 = 74, 430/7 < x2, x2 < 62}
Region6
{410/7 < x2, x1 < 76, x2 < 414/7, -(7/4)*x2+357/2 < x1}, {414/7 <= x2, 75 < x1, x1 < 76, x2 < 60}, {x1 = 75, 414/7 < x2, x2 < 60}
Region7
{404/7 < x2, x1 < 76, x2 < 58, -(7/4)*x2+177 < x1}
Region8
{408/7 < x2, x1 < 75, x2 < 412/7, -(7/4)*x2+177 < x1}, {412/7 <= x2, x2 <= 414/7, 74 < x1, x1 < 75}, {x1 = 74, x2 <= 418/7, 412/7 < x2}, {74 < x1, 414/7 < x2, x1 < -(7/4)*x2+357/2, x2 < 418/7}, {x1 = -(7/4)*x2+357/2, 414/7 < x2, x2 < 418/7}
Region9
{x2 = 60, 147/2 < x1, x1 < 74}, {60 < x2, x1 < 74, x2 < 426/7, -(7/4)*x2+357/2 < x1}, {426/7 <= x2, x2 <= 430/7, 72 < x1, x1 < 74}, {x1 = 72, 426/7 < x2, x2 < 62}, {72 < x1, 430/7 < x2, x1 < -(7/4)*x2+363/2, x2 < 62}, {x1 = -(7/4)*x2+363/2, 430/7 < x2, x2 < 62}
Region10
{58 <= x2, 72 <= x1, x2 <= 412/7, x1 < 74}, {x1 = 72, 412/7 < x2, x2 < 60}, {72 < x1, 412/7 < x2, x1 < -(7/4)*x2+177, x2 < 60}, {x1 = -(7/4)*x2+177, 412/7 < x2, x2 < 60}
Region11
{56 <= x2, x2 <= 402/7, 70 < x1, x1 < 72}, {x1 = 70, 56 < x2, x2 < 58}, {70 < x1, 402/7 < x2, x1 < 345/2-(7/4)*x2, x2 < 58}, {x1 = 345/2-(7/4)*x2, 402/7 < x2, x2 < 58}
Region12
{x2 = 54, 70 <= x1, x1 < 72}, {x1 = 70, 54 < x2, x2 < 55}, {54 < x2, 70 < x1, x1 < 333/2-(7/4)*x2, x2 < 55}, {x1 = 333/2-(7/4)*x2, 54 < x2, x2 < 55}
Region13
{x2 = 56, 68 <= x1, x1 <= 137/2}, {x1 = 68, x2 <= 394/7, 56 < x2}, {56 < x2, 68 < x1, x1 < 333/2-(7/4)*x2, x2 < 394/7}, {x1 = 333/2-(7/4)*x2, 56 < x2, x2 < 394/7}

The correct 3d plot:

@Preben Alsholm 

To obtain the 1D version I have pasted the Int in a 2D execution group and then lprint(%).
BTW, the exact and approx value of the integral are computed below.
 

@Carl Love 

OP's function written in a natural and simplified form is

f:=piecewise(0<-w*x*y+u*x and -w*x*y+u*x<1,1)*piecewise(0<-w*x*y+v*y and-w*x*y+v*y<1,1) ;

Now,

int(f,  [x=0..1,y=0..1,u=0..1,v=0..1,w=0..1]);
       7/12

Converting to Heaviside
g:=convert(f,Heaviside);
        g := Heaviside(-w*x*y+u*x)*Heaviside(-w*x*y+v*y)*Heaviside(w*x*y-u*x+1)*Heaviside(w*x*y-v*y+1)
int(g,  [x=0..1,y=0..1,u=0..1,v=0..1,w=0..1]);
    (interrupted)
but ...

int(g,  [u=0..1,v=0..1,w=0..1,x=0..1,y=0..1]);   # ok!
                               7/12

I wanted to check manually the result.
I was surprised to find that Maple is not able to solve the system

[0 < -w*x*y+u*x, -w*x*y+u*x < 1, 0 < -w*x*y+v*y, -w*x*y+v*y < 1, 0 < x, x < 1, 0 < y, y < 1, 0 < u, u < 1, 0 < v, v < 1, 0 < w, w < 1]

using solve, or SolveTools:-SemiAlgebraic

even if it's easy to see that the solution is:

0<x<1, 0<y<1, 0<w<1, w*y<u<1, w*x<v<1

so the integral is indeed
int((1-w*x)*(1-w*y), x=0..1, y=0..1, w=0..1) = 7/12.

@mmcdara 

It's about the user, not the laptop ... :-)

@mmcdara 

Have you tried the code?

discont was added just for a correct plot.

It seems that you actually do not want an answer. In this case you can simply ignore mine.

kernelopts(version);
   Maple 2018.1, X86 64 WINDOWS, Jun 8 2018, Build ID 1321769

bin.mw

@Klausklabauter 

So, LDLt is actually implemented in

Student:-NumericalAnalysis:-MatrixDecomposition(A, 'method'='LDLt')

Note however that it is too slow (compared with Cholesly for floats) for large matrices.

@Ahmed111 

For p:=1; q:=1; lambda:=2; i:=1  the system has only the trivial solution.
Maple says u1=u2=0  for i = I  too. If you want to check this, try to use the method from the provided worksheet.

@ThU 

Yes, this is what MakeInert does (automatically).

@mmcdara 

I just see that the two parameters are switched compared with some (all?)  standard references.
Have you noticed a bug related to this? As long as Maple is consistent, it seems to be OK.

Even in mathematics the notations are not standardized.
E.g. for some authors N = {0,1,2,...}  but for others N = {1,2,...}.

A much more annoying situation in Maple is that the conventions are not the same everywhere:

spherical coordinates in plot3d:
x = u cos(v) sin(w)
y = u sin(v) sin(w)
z = u cos(w)

spherical coordinates in  VectorCalculus:
x = u cos(w) sin(v)
y = u sin(v) sin(w)
z = u cos(v)

i.e. phi and theta are switched!

@Carl Love 

I agree for frem but not for trunc  which satisfies the identity  frac(x)+trunc(x) = x.
Being built in is not necessarily relevant; probably frac(x) should be more often replaced (internally)  by x - trunc(x).

@Markiyan Hirnyk 
Don't you also think that the last sentence could have been omitted?

@Carl Love 

The non-analytic functions frac, trunc, floor etc do not like calculus.
They seem to have incomplete implementations.
frac(x)+trunc(x)  is not simplified to x  and
simplify(floor(u)) assuming u<4,u>3;  remains unsimplified.
 

Edit. Using trunc, Maple gives the corect result.

s := x -> min(frac(x),1-frac(x)):
b := x -> sum(s(2^n*x)/2^n, n=0..4):
L:=Limit( ( b(x)-b(181/20) )/(x-181/20), x=181/20 ):
value(L);
                               0
LL:=eval(L, frac=(t -> t - trunc(t)) ):
value(LL);
                               3

@tomleslie 

Your version is nice too.
For my version two conditions were imposed:
(1)  Each quadrilateral (black or white) is a rhombus.
(2) The first 32 black rhombuses (the fat ones) are actually squares.
These two conditions determine uniquely the tiling (except the circle in the middle, of course), and it seems that the original tiling satisfies (1)+(2).