I suppose at the time this book was written, 100 years ago, that Steiners worlds seemed revelatory. But for anyone with exposure to metaphysical concepts like higher dimensions this book does not deliver.
This book was not written by Steiner but is a collection of his lectures on the subject of The Fourth Dimension. And the publishers choice of including the words sacred geometry and alchemy is somewhat misleading. Steiner is a controversial figure because much of what he talked about is hard to follow. In this book Steiner spends many lectures in which a great deal of the current lecture/chapter repeats the previous content.
In this book much of what he talks about does not agree with my own intuitive perceptions of higher realms or with what I have heard from others experiences. For instance, Steiner followed a train of thought that said that in order to go past any dimension, especially the third, that you need a kind of dimensional mirror to shortcut the limitations of the current dimension and add the next dimension. And in the case of getting past the the third dimension this lead Steiner to state that numbers in the fourth dimension (Astral realm) must be read backwards. If you see the number 782 in the Astral realm that must really be the number 287. Now keep in mind that while explaining all of his ideas about the fourth dimension Steiner never really tells you how to perceive it. There are no practices or techniques to use. Later in the book Steiner seems to either clarify or reverse himself when he says that the fourth dimension is time, something most people in metaphysics would be more likely to agree with rather than reversed numbers.
Steiner does go on to imply that animals seen outside of you in the Astral realm are really symbolic or occult representations of your internal state. This much agrees with shamanic views of power animals or spirit helpers.
The majority of the book is a series of question and answer topics. This area covers a wide range and is not organized into topics so it's a bit like a fishing expedition.
For me this was the second Stiener book I have tried to read the first being Egyptian Myths and Mysteries. While the Egyptian book had much more information or facts I was still left thinking 'so what' or 'how can that be proven' to many things Steiner claimed. I also have in my stack of books Biodynamics by Steiner and from flipping through that book it seems that will be a more meaningful book to me as I have an interest in sustainable agriculture and the spiritual forces involved.
From what I can tell thus far Steiner was greatly influenced by Goethe. So any future attempts I make into Steiner land will include either the book(s) Stiener has written about Geothe or other books by/about Goethe.
Finally, from my overall experience of Steiner, I believe he was a genuine man, sort of a prophet of his time. But as a civilzation I believe much of what he has said we have reached at least by a great deal of the metaphysical community consciously and by many of more of us unconsciously. So for most of my metaphysical peers many of the Steiner books will be difficult reading seemingly sparse of usefull information.
My final concern is that when people progress along the spiritual path there is no single version of the reality, short of the ultimate truth, that people all journey with. Though advanced souls may experience the astral plane they each do so from a unique viewpoint. So when people take Steiners views and make something like the Waldorf educational system, I take pause with some concern. I believe that there is some value in Waldorf, but like the body of Steiners works I have read thus far, I have to say that some of what I hear sounds squirrely.
So to my fellow spiritual seekers I say that I have realized that I am lookiing for spiritual nuggets to add to my understanding, not for someone to follow or model my walk on. And in the case of Steiner the nuggets in his field seems to be sparse. Be well.
P.S. If you were looking for an introduction to Sacred Geometry, the Lawlor book is at least four dimensions better.