Dear mathematicians

I have a question regarding the intersection of three dimensional hyperplanes in R^4. I know that the intersection of two planes in R^2 is a line. So the intersection of a finite number of planes is a line and we can thought of the planes as a pages of an open book.

How can we define the intersection of three-dim hyperplanes in R^4. Is it true that the intersection of finite number of hyperplanes is a common plane between them, if so how can we visualise and sketch this.

Thank you in advance