The Answer is:
From the White House to the Trevi Fountain
Both have columns along the facade.
I have sought advice from secret sympathisers
Advice columns (like ask amy type stuff). A fifth column is any group of people who undermine a larger group.
Stalactites met stalagmites and orchids bloomed
When they meet it forms a continuous line top to bottom or a column. The column, or technically the gynostemium, is a reproductive structure that can be found in several plant families: Aristolochiaceae, Orchidaceae, and Stylidiaceae. wikipedia
There were falling blocks and Napoleonic Attacks
Falling blocks refers to Columns a match-three puzzle video game, first created in 1989 Originally guessed tetris woops. . A military column is a formation of soldiers marching together in one or more files in which the file is significantly longer than the width of ranks in the formation. Originating in the french revolutionary war.
PSE pages held the answer
There are columns the pages are organized by like the related, hot network question column.
Juggling spreadsheets and matrices
Both Spreadsheets ad matrices have columns. In toss juggling, columns, also known as One-up Two-up, is a juggling trick or pattern where the balls are thrown upwards without any sideways motion. The simplest version involves having three balls, with two going up simultaneously on either side, followed by one going up in the middle.
Distilling it all to a cluster of neurons
there are small areas (columns) in which the neurons tend to have the same receptive field. A fractionating column or fractionation column is an essential item used in distillation of liquid mixtures so as to separate the mixture into its component parts, or fractions, based on the differences in volatilities. (distillation).