By Lee C. F. Sallows

This leading edge paintings via a British engineer initiates a Copernican revolution in our realizing of magic squares via changing numbers with two-dimensional types. the result's no longer in basic terms a singular type of geometrical magic sq. yet a revelation that conventional magic squares are actually larger noticeable because the one-dimensional example of this self-same *geometrical* activity.

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**Extra info for Geometric Magic Squares: A Challenging New Twist Using Colored Shapes Instead of Numbers**

**Example text**

Xn ) of a point X with respect to R and the coordinates (x1 , . . 7). To ﬁnd this relationship we put −−→ PX = n xi ei , i=1 −−→ P X= n xj vj , j=1 −−→ PP = n bi ei , i=1 18 1. 7. Coordinates of X with respect to R and R n vj = aij ei . 1) i,j=1 and also −−→ −−→ −−→ P X = P P + PX = − n n bi ei + i=1 xi ei . 2) one obtains n xi = xj aij + bi , i = 1, . . , n. j=1 These n equations can be written as ⎞ ⎛ x1 a11 ⎜ .. ⎟ ⎜ .. ⎝ . ⎠=⎝ . ⎛ xn an1 ... . ⎞⎛ ⎞ ⎛ ⎞ x1 a1n b1 .. ⎟ ⎜ .. ⎟ + ⎜ .. ⎟ .

15), one obtains λμν = 1, and this completes the proof. Note that in the statement of Menelaus’ theorem it is necessary that P = A, B, Q = B, C, R = C, A, that is, the straight line that cuts the sides of the triangle does not contain any vertex. 43, page 45. 36 In the axiomatic development of Euclidean Plane Geometry it is not possible to prove directly that if a straight line intersects one side of a triangle and misses the three vertices, then it must intersect one of the other two sides. This must be imposed as an axiom, as was observed by Moritz Pasch and included in D.

Find the equations of the change of coordinates between the aﬃne frames R1 and R2 of the aﬃne space R3 given by R1 = {(1, 1, 1); (e3 − e1 , e1 + e3 , e1 + e2 )}, R2 = {(0, 1, 1); (e1 , 2e2 − e3 , 3e1 − e3 )}. 35. Consider the aﬃne frame R of the aﬃne space R3 given in the canonical aﬃne frame by R = {(1, 1, 1); ((1, 1, 1), (0, 1, 0), (2, 1, 0))}. What are the coordinates of the point (0, 0, 0) ∈ R3 in this new aﬃne frame? And those of the point (1, 1, 1) ∈ R3 ? Is there a point with the same coordinates in R as in the canonical aﬃne frame C?