**The Fano Plane —**

"A *balanced incomplete block design* , or BIBD

with parameters *b *, *v *, *r *, *k *, and *λ * is an arrangement

of *b* blocks, taken from a set of *v* objects (known

for historical reasons as *varieties* ), such that every

variety appears in exactly *r* blocks, every block

contains exactly *k* varieties, and every pair of

varieties appears together in exactly *λ* blocks.

Such an arrangement is also called a

(*b *, *v* , *r* , *k* , *λ* ) design. Thus, (7, 3, 1) [the Fano plane]

is a (7, 7, 3, 3, 1) design."

— Ezra Brown, "The Many Names of (7, 3, 1),"

Mathematics Magazine , Vol. 75, No. 2, April 2002

W. Cherowitzo uses the notation (v, b, r, k, λ) instead of

Brown's (*b* , *v* , *r* , *k* , *λ* ). Cherowitzo has described,

without mentioning its close connection with the

Fano-plane design, the following —

"the (8,14,7,4,3)-design on the set

X = {1,2,3,4,5,6,7,8} with blocks:

{1,3,7,8} {1,2,4,8} {2,3,5,8} {3,4,6,8} {4,5,7,8}

{1,5,6,8} {2,6,7,8} {1,2,3,6} {1,2,5,7} {1,3,4,5}

{1,4,6,7} {2,3,4,7} {2,4,5,6} {3,5,6,7}."

We can arrange these 14 blocks in complementary pairs:

{1,2,3,6} {4,5,7,8}

{1,2,4,8} {3,5,6,7}

{1,2,5,7} {3,4,6,8}

{1,3,4,5} {2,6,7,8}

{1,3,7,8} {2,4,5,6}

{1,4,6,7} {2,3,5,8}

{1,5,6,8} {2,3,4,7}.

These pairs correspond to the seven natural slicings

of the following eightfold cube —

Another representation of these seven natural slicings —

These seven slicings represent the seven

planes through the origin in the vector

3-space over the two-element field GF(2).

In a standard construction, these seven

*planes *provide one way of defining the

seven projective *lines* of the Fano plane.

A more colorful illustration —