I've been on a search for a while now but I can't figure out the term for this design element. I'm not even really sure what to look up. I tried going through terms associated with typography but no dice. You find it used a lot in vintage texts like newspaper ads and wanted posters where identical symbols would be used on either side of a large bold text to create emphasis. Sometimes the symbols would be stars, sometimes dots, and sometimes these sharp darts.
Flourishes, ornaments, fleurons, dingbats, embellishments.
Basically any term which describes additional, possibly unnecessary, ornamental, items. There's no real specific typography term.
(Except perhaps dingbat which designates a font file containing non-alphanumeric glyhs.
And fleuron generally refers to more "flowery" ornamental glyphs.)
In the sample image, the rules above and below are as much flourishes, ornaments, embellishments, as the diamonds.