MathOverflow

In the proceedings of a 1976 conference on abelian group theory, Edward R. Fisher presented a paper titled "Abelian structures I." Besides the suggestive number in the title, it contains the following paragraph:
In a sequel we intend to discuss criteria for elementary equivalences and elementary embeddings, a Lowenheim-Skolem theorem for $\infty\eta$-substructures of injectives, yielding proofs of the existence of homogeneous-universal and saturated models in...

Question 0:How do injective envelopes work in constructive mathematics?
For example,
Question 1: How strong is it to assert internally that there are enough injectives (in the category of sets, say)? (Externally, every topos has enough injectives thanks to the subobject classifier.)...

An article in theNotices of the AMS, Volume 61, Issue 10, 2014(PDF download link),on Khot's Unique Games Conjecture, says this:
Another group ... found a shape that in a certain sense lies halfway between a square and a circle (though in many more than two dimensions). Like a square, copies can be placed next to each other horizontally and vertically to fill a whole space without...

What is known about the area of the symmetric Pythagorean tree? (Closed unit square as base, area of enclosed triangles not included.) The problem in calculating the area is that squares start to overlap from the sixth iteration onwards.
Without much effort it can be computed that height is 4 and breadth is 6, so an upper bound for the area is 24. Wikipedia reports that lower and upper bounds can with some effort be tightened to 5 and 18, respectively. The accompanying Wikipedia talk...