The following text is taken from the a book about linear programming that I'm reading:
A graphical illustration is useful for understanding the notions and procedures of linear programming, but as a computational method it is worthless. Sometimes it may even be misleading, since objects in high dimension may behave in a way quite different from what the intuition gained in the plane or in three-dimensional space suggests.
I don't know what the author is insinuating at by saying that the object's behavior is different from the intuition since my experience working with objects in higher dimension was always gained from lower dimension.
So, is there some concrete example illustrating this statement?
You are missing what the Author is trying to convey.
(1) A graphical illustration is useful for understanding (by humans) the notions and procedures of linear programming, but as a computational method (by computers) it is worthless.
(2) Sometimes it may (not "will") even be misleading, since objects in high dimension may (not "will") behave in a way quite different from what the intuition gained in the plane or in three-dimensional space suggests.
Nobody except the Author can tell what thoughts led to that claim.
I will try to give Examples for these from my thoughts.
EXAMPLES :
(1A) There are Pairs of Curves which seem to intersect once , when viewed with DESMOS or Wolfram. When analysed with Calculus or other tools , it is revealed that there are 3 intersections.
It might even turn out that there is some very tiny sinusoidal nature at the central intersection , giving even more intersections.
Such things will not be visible with graphing methods.
We still know that there is at least 1 intersection , when we view the graphing output. Hence it is useful to know.
It is worthless when we try to get the actual intersection value.
(1B) To make the graph , the functions were evaluated at hundreds or even thousands of Points. It is computationally wasteful , giving us not much insight into the nature.
Instead , if we had used some calculus or other techniques , we might have got closed form values for the Intersection Points , with much less computational effort.
More-over , we will know much more , like number of Solutions , whether we have tangents , what if Parameters to the functions were varied , what is the range of the Intersection Points , etc
We have to give input to various numerical algorithms. These inputs are generally in symbolic or numeric form , not in graphical form. Eg When we want to get Extrema values or Intersection Points , then we give the Equations of lines / circles / functions. It is worthless to give Images of lines , circles & functions.
(2A) Consider 1D line.
Let us make some arrow from $1$ to $2$. Can we reverse the arrow , without leaving that Domain ? NO
Let the arrow be in 2D Plane from $(1,0)$ to $(2,0)$. Can we reverse the arrow , without leaving that Domain ? YES
We can rotate the arrow around and then shift it.
Intuition in 1D is wrong in 2D.
(2B) Consider a Point $x_1$ in 1D moving from $0$ to $3$.
Does this Point $x_1$ have to go through $2$ ? YES
Consider a Point $x_2$ in 2D moving from $(0,0)$ to $(3,0)$.
Does this Point $x_2$ have to go through $(2,0)$ ? NO
It can move up , then right , then down. It can avoid $(2,0)$.
(2C) Update our intuition : Does this Point $x_2$ have to go through the circle $x^2+y^2=1$ ? YES
Moving up in Dimensions , we have to update out intuition suitably.
(3) Consider the Purple Curve enclosing a hole in the region with 2 holes in 2D.
Can we make "Continuous Deformations" such that the Purple Curve contains 2 holes ? NO
Suppose that the region was 3D.
Can we make "Continuous Deformations" such that the Purple Curve contains 2 holes ? YES !!
There are many Examples where our Intuition helps us when we move to higher Dimensions.
Yet , there are many other Examples where our Intuition fails us when we move to higher Dimensions.
Best we can do is to update our Intuition along the way.
SUMMARY :
1D 2D 3D Images & Illustrations & graphs may help us , though these might not help in computations. Occasionally , these might mislead too , when we encounter some new Situation in higher Dimensions.
DISCLAIMER :
This Question is Opinion Based & there can be no "comprehensive" closed form answer.