I S O M O RPH IC MA C H IN E S
Study of a Black Box can thus give the experimenter infor-
Mation up to a certain amount; and, if the inputs and outputs are
Given, cannot possibly be made to give more. How much informa-
Tion will be discussed in S. 13/15 (especially its last Ex.). Here it
Is sufficient if we notice that the canonical representation speci-
Fies or identifies the mechanism “up to an isomorphism”.
Isomorphic” means, roughly, “similar in pattern”. It is a con-
Cept of the widest range and of the utmost importance to all who
Would treat accurately of matters in which “pattern” plays a part.
Let us consider first a few examples merely to illustrate the basic
Ideas.
A photographic negative and the print from it are, so far as the
Pattern of the picture is concerned, isomorphic. Squares in the
Negative appear as squares in the print; circles appear as circles;
Parallel lines in the one stay as parallel lines in the other. Thus cer-
Tain relations between the parts within the negative appear as the
Same relations in the print, though the appearances so far as
Brightness is concerned are different, exactly opposite in fact.
Thus the operation of changing from negative to print leaves these
Relations unaltered (compare S.5/2).
A map and the countryside that it represents are isomorphic (if
the map is accurate!). Relationships in the country, such as that
Towns A, B and C form an equilateral triangle, occur unchanged
On the map, where the representative dots for A, B and C also
Form an equilateral triangle.
The patterns need not be visual. If a stone is thrown vertically
Upwards with an initial velocity of 50 ft. per second, there is an
Isomorphism between the set of points in the air such that at time
T the stone was h feet up and the set of those points on a graph that
Satisfy the equation
y = 50x – 16x2.
The lines along which air flows (at sub-sonic speeds) past an
Aerofoil form a pattern that is identical with the lines along which
Electric current flows in a conducting liquid past a non- conductor
Of the same shape as the aerofoil. The two patterns are the same,
Though the physical bases are different.
Another isomorphism is worth consideration in more detail.
Fig. 6/8/1 shows two dynamic systems, each with an input and an
Output. In the upper one, the left-hand axle I is the input; it can be
Rotated to any position, shown on the dial u. It is connected
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Through a spring S to a heavy wheel M, which is rigidly connected
To the output shaft O. O’s degree of rotation is shown on the dial
V, which is its output. The wheel dips into a trough with liquid F
Which applies a frictional force to the wheel, proportional to the
Wheel’s velocity. If now, starting from given conditions, the input
U is taken through some sequence of values, so will the output v
Pass through some determinate sequence of values, the particular
Sequence depending on v’s initial value, on v’s rate of change at
That moment, and on the sequence used for the input at u.
Fig. 6/8/1
The lower system is electrical. Its input is a potentiometer, or
Other device, J, that emits the voltage shown on the scale x. In
Series are an inductance L, a resistance R, and a capacitance C. P
Is a current meter (such as is used in domestic supplies) recording
The sum of the currents that have passed through it. The sum is
Shown on the scale y, which is its output.
If now the values of L, R and C are adjusted to match the stiff-
Ness of the spring, inertia of the wheel, and friction at F (though
Not respectively), then the two systems can show a remarkable
Functional identity. Let them both start from rest. Apply any
Input-sequence of values at u, however long and arbitrary, and get
An output sequence at v, of equal length: if the same sequence of
Values is given at x, the output at y will be identical, along its
Whole length with that at v. Try another input sequence to u and
Record what appears at v: the same input given to x will result in
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A N I N T R O D UC T I O N T O C Y B E R NE T I C S
TH E BL AC K B O X
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