TITLE: Are Rigid Unit-Distance Graphs Algebraic?
QUESTION [5 upvotes]: Connect vertices (0,0) and (1,0) with a unit-distance edge. Add more unit distance edges so that the resulting graph has structural rigidity.   
Are all vertices necessarily algebraic?  
EDIT:  Here's an example of a hard case, the Fish graph by Hochberg–O'Donnell, the smallest known triangle-free 4-chromatic unit-distance graph.  

Points $(15, 17, 19, 21, 23)$ are at 
$((-0.824076,0.566479), (0.324076,-1.06648), (-0.199583,-0.214551),(-0.632394,-0.774647), (-0.632394,-0.774647))$   
the real roots of index $((2,4),(3,1),(3,2),(3,2),(2,5))$ for the following polynomials:
$(-319-216 x+1824 x^2+576 x^3-3072 x^4+2048 x^6, \\ 
 13+456 x-1728 x^2-1600 x^3+4608 x^4+6144 x^5+2048 x^6, \\  
 -143-1536 x-4320 x^2+2112 x^3+20736 x^4+27648 x^5+18432 x^6, \\  64573-91836 x-3233592 x^2-7450560 x^3+23213376 x^4+135454464 x^5+203751936 x^6-48771072 x^7-433963008 x^8-331776000 x^9+127401984 x^{10}+254803968 x^{11}+84934656 x^{12}, \\  -10487+51300 x+418536 x^2-965952 x^3-4849344 x^4-3697920 x^5+40020480 x^6+56733696 x^7-135364608 x^8-199065600 x^9+127401984 x^{10}+254803968 x^{11}+84934656 x^{12})$ 
All vertices are algebraic numbers. Considering the amount of wizardry I needed to find this exact solution, I believe this is non-constructible.  As a side question, why are all these coordinates using the same polynomial?

REPLY [2 votes]: Yes. The co-ordinates of each vertex in such a graph can be defined in the theory of real-closed fields, and are hence algebraic by the Tarski–Seidenberg theorem. Unit distance constraints can be expressed as equalities, and the definition of rigidity gives you a $\delta$ which can be assumed to be rational and then used in a real inequality to define the condition of lying in a particular open ball of radius $\delta.$
The converse that any positive algebraic number can appear as a distance in a rigid unit-distance graph was proved by Hiroshi Maehara, Distances in a rigid unit-distance graph in the plane,  https://www.sciencedirect.com/science/article/pii/0166218X9190070D (The first result I got for "rigid unit distance algebraic" on Bing, after this thread.)