By Howard Eves

From the book's preface:

Since writing the preface of the 1st variation of this paintings, the gloomy plight there defined of starting collegiate geometry has brightened significantly. The pendulum turns out certainly to be swinging again and a goodly quantity of fine textual fabric is showing.

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Am. Math. Soc. 62, 114–192 (1947) 28. H. Federer, Geometric Measure Theory (Springer, Berlin/Heidelberg/New York, 1969). pbk. reprint 1996 29. A. Ferguson, J. Fraser, T. Sahlsten, Scaling scenery of . m; n/ invariant measures. Adv. Math. 268, 564–602 (2015) 30. A. Ferguson, T. Jordan, P. Shmerkin, The Hausdorff dimension of the projections of self-affine carpets. Fund. Math. 209, 193–213 (2010) 31. H. Furstenberg, Ergodic fractal measures and dimension conservation. Ergod. Theory Dyn. Syst. 28, 405–422 (2008) 32.

Fenn. A Math. 1, 387–392 (1975) 52. R. Kenyon, Projecting the one-dimensional Sierpinski gasket. Isr. J. Math. 97, 221–238 (1997) 53. D. Khoshnevisan, Y. Xiao, Packing-dimension profiles and fractional Brownian motion. Math. Proc. Camb. Philos. Soc. 145, 145–213 (2008) 54. F. Ledrappier, E. Lindenstrauss, On the projections of measures invariant under the geodesic flow. IMRN 9, 511–526 (2003) 55. M. Leikas, Packing dimensions, transversal mappings and geodesic flows. Ann. Acad. Sci. Fenn. A Math.

A0 / 2 f0; 1g for P-almost all !. If P! P! / Ä d, and if P! P! / Ä k. f! W P! P! d k/. But this contradicts the choice of ı. Thus the claim holds. 3. 4]. 5 If k 2 f1; : : : ; d 1g, k < s Ä d, and 0 < ˛ Ä 1, then there exists a Radon measure on Rd with dim. d; k; ˛/; for -almost all x 2 Rd . d; k; ˛/ does not depend on the choice of V. d; k/. Since P is a convex combination of two fractal distributions, it is a fractal distribution. 9. 2, we see that is exact-dimensional and dim. P/ D s d kÁ k D s: k s d k dC 1 k The goal is to verify that has the claimed properties.