We could of course just fall back to punishing people who do beacon griefing, they are quite easy to track 
I don’t really think a lot of people would use extra expenive beacons to build roads, it seems to me that roads either needs to be somewhat cheap to make or else they will not be made at all.
Alright… it seems people doesnt understand what i mean.
you say “if its more expensive people wont use it”
Lets just assume that you want a 4x1x128 road. Yeah? if we used “Normal beacons” you would get 8x8x8 which means to cover the 128 which matters you would need to put down Sixteen beacons. while if you use the infrastructure beacon you get the price of only Three beacons, if we use three times the price BUT If people try to use them for 8x8x8 they would also have to pay for Three Beacons even though One building beacon could cover it.
I also want to point out that this system kinda does get more expensive if you try to use it in a grief case, as each 8x8x8 plot would use one of your permanent alotment. Wall griefing would be either very costly, or no taller than 8, maybe 16 blocks high. It would be exponentially more expensive the larger the diameter of the target, where as the XxYxZ system could wall off larger targets with a minimum cost.
Special beacons for roads/ways are ok, but for gardens you should use normal beacons, cuz having a nice garden is like having a big house, so pay for your decadence … and also: Gardens are not as high as buildings, so you would need less beacons for them (for the same size).
But I think there shoud be a way to help out here: Placing a beacon costs you a “normal beacon” at first … if you size it up each plot should cost less, so for example: …
1 beacon = 1
2x2 beacons = 2
3x3 beacons = 3
4x4 beacons = 4
5x5 beacons = 5
for each extra 5 beacons = +1
on this way it would be cheaper until a set size and then grows on continually … of cause each crafted beacon should cost the same mats as normal, cuz mats are not bound to a maximum 
Welp. precisely same problem with your system, nothing prevents a dude from making a wall of dirtblocks.
Either i dont understand what you mean or there is something severely wrong with the math.
Well, to circle a single beacon, you would need 8 more. To circle a 2x2 of beacons, you would need 12. Bigger targets- more of the griefers alotment, less potential for damage.
Sorry, i have no clue what you are trying to say, can you rephrase it?
the larger you build a single beacon/beaconed area the less of your maximum limit is “consumed” … of cause this should not end in a unlimited reducing of the costs per “ring” or “length” added, cuz that would lead to the question “hmm, 100 single beacons or one beaconed area with a diameter of 200 beacons?” ^^ …
So in my example the single added beacons get less explansive untill you have reached 5x5 (25 beacons) at which point each added beacon costs a fixed value.
also, this could be used for ways … but that would also lead to a possibility of greifing through just building a loooooong wall of beacons around or through settlements 
…
may be my math is not perfect or correct, but the discribtion now should at least explain my idea to you ^^
It still makes zero sense, you know that a Beacon is 8x8x8 blocks right? do you mean blocks’?
nope, I ment beacons placed next to each other to cover a greater area with “one” beacon
Oohkay, imagine beacons as they line up on a grid. If I were trying to wall grief a players beacon, I would want to completely surround their beacon, yes? So if their beacon is in the middle, I would need 8 beacons to go around the outside. G= griefers beacon, X= players beacon.
GGG
GXG
GGG
It takes 8 G to wall in just 1 X. If my base were bigger, though…
GGGG
GXXG
GXXG
GGGG
It costs the griefers more to wall me in. Since everybody has a set alotment of these landscape beacons, griefers can only do a set amount of damage with this system (before they are banned, that is!)
Also, the base cost to grief even a single small beacon is very steep at 8-1.
If you made the alotment of beacons start low and go up based on some factor of good game play (rep, time played, achievements, what ever) then you could also prevent griefers from having acess to enough of these landscape beacons to do any notable damage from the start.
So? people can grief with whatever system they want, if people grief, report them, I think Birne’s idea is the best and has also been mentioned, if you place a “building beacon” no beacon can be placed in a 3 block area around the beacon, that quickly expands how many beacons are needed to grief. Again your system wouldnt differ from mine in that sense, the only different would be which block you could build a wall beacon with.
If people grief, report them. simple.
Just if it isnt clear, i dont disagree with you, this system of freedom DEFINITELY makes it easier to grief. the math checks out after all. In the end everything has a downside, is the upside of infrastructual freedom worth the increased ease of which people can grief.
Not having symmetry bothers me on an extremely deep level. Call me a sucker, but I think it’s important for these types of games 
I like the idea of “no beacon next to other (non friended) beacons”, this just enables me to choose to modify the landscape in that region of my own beacon, or even extend the area covered in the direction of my own choosing. I think if players chose to use it for grief, it would be easy to report it as such, no matter what system used.
The only “Problem” or “Weird thing” i can see with the system is if forexample we use the number 3 blocks out.
lets say i place a beacon and another person cant place within those 3 blocks, but he places it right outside my three blocks, then my three blocks and his three blocks overlap, then what?
It would make sense to say you cant place within 8x8x8 distance around the beacon, but im afraid that might be too much and also abuseable, however if you have less it sort off ruins the point.
I don’t really feel that 8 is too far. Feels kinda “right” to me.
Agree! It always drove me up a wall when playing SimCity 
This!
In regards of roads:
I think roads (in large scales) should be a feature exclusive to guilds. When you found a guild you get a giant beacon for your guildhall/city/whatever and a bunch of the normal 8x8x8 ones that have to be connected to the main beacon and can be used for roads or other stuff. If you still want a road at your house you can use some of your personal beacons but I really don´t think that every player should have a set of dedicated beacons exclusive for roads. It would kinda ruin the sandbox/exploration aspect of the game if you would encounter a plastered road too frequently.
That’s a good counterbalance and I like it.