Snowman Math

Orianna2000

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Okay, this is a bit of an oddball question.

I've two characters who go up to the roof of their four-story tall apartment building to build a snowman, because the snow on the street is all slushy and gross. I want one of the characters, who is highly analytical and logical, to calculate how big of a snowman they can theoretically make, based on the volume of snow on the roof.

I'm utter rubbish with math, but looking at a not-to-scale map of the apartment, I'm guessing the square footage of the roof would be about 900 square feet. (It's in London, but I haven't converted all the numbers yet. Do they use square meters for house sizing in the UK?) The snow is currently about 5" deep, but this can be changed, if need be.

I've written that they could, in theory, build a snowman with a base that's 2 meters in diameter and 4 1/2 meters tall, but that's just random numbers that sound good, based on typical snowman ratios. I've no idea if it works with the size of the roof and depth of snow. Basically, I want the proposed snowman to be ridiculously huge, so the other character can veto it in favor of a normal snowman. So if the roof is 900 square feet, how deep does the snow need to be in order to make a giant snowman? (Note: The snow depth and the size of the snowman are variables that can be changed, but the roof size needs to stay at 900 square feet.)

Please, please don't respond by telling me how to do the calculations myself. I have dyscalculia, so math is my Moriarty. Someday, we'll die with our hands wrapped around each other's throats.
 

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My neighbors in Arizona did the "snowman on the roof" thing one year, when we got 4-5 inches of snow. It stuck to lawns and roofs, but they had gravel in their yard, which melted the snow quickly, so the roof was their only option.

Their house was, ballpark, 1,100 square feet. Pitched roof, which would have more surface area than a flat one. The final snowman was about 3 1/2 feet high and they used most of what they had. I'd think they'd need some super-deep snow to make a giant snowman like what you're describing.

Re: the square footage, That's reallly tall and skinny for a stand-alone structure, but not unusual for buildings that are squished up against their neighbors. If it were the latter, then they might have access to additional snow on the next roofs over.
 
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cornflake

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Okay, this is a bit of an oddball question.

I've two characters who go up to the roof of their four-story tall apartment building to build a snowman, because the snow on the street is all slushy and gross. I want one of the characters, who is highly analytical and logical, to calculate how big of a snowman they can theoretically make, based on the volume of snow on the roof.

I'm utter rubbish with math, but looking at a not-to-scale map of the apartment, I'm guessing the square footage of the roof would be about 900 square feet. (It's in London, but I haven't converted all the numbers yet. Do they use square meters for house sizing in the UK?) The snow is currently about 5" deep, but this can be changed, if need be.

I've written that they could, in theory, build a snowman with a base that's 2 meters in diameter and 4 1/2 meters tall, but that's just random numbers that sound good, based on typical snowman ratios. I've no idea if it works with the size of the roof and depth of snow. Basically, I want the proposed snowman to be ridiculously huge, so the other character can veto it in favor of a normal snowman. So if the roof is 900 square feet, how deep does the snow need to be in order to make a giant snowman? (Note: The snow depth and the size of the snowman are variables that can be changed, but the roof size needs to stay at 900 square feet.)

Please, please don't respond by telling me how to do the calculations myself. I have dyscalculia, so math is my Moriarty. Someday, we'll die with our hands wrapped around each other's throats.

Yes, they use meters, so that'd theoretically be like 300ish meters, but kind of depends on the dimensions, which you haven't given.

The problem here isn't the math, happy to help with the math, the problem is kind of twofold: I've got no idea how much snow would be required to build a snowman with a base 2 metres in diameter, and if it's 4.5 m tall (I assume you mean the entire thing), the three balls will be different sizes. Additionally, snow matters! Fluffy snow takes up room but doesn't pack, wet snow packs but doesn't take up as much volumetric space, etc.

That said, I doubt anyone will be checking your math, so I think you're probably fine winging it to an extent.

The volume of the bottom ball, without taking snow density into account, would be like 4.2m^3 (cubic metres).

The main issue with all of this for me is I'm afraid you're caving in the roof. If I were reading it, that'd be what I'd be thinking -- snow weighs a lot, especially packable snow, and packing a giant-ass snowman's worth onto a flat roof seems like a terrible idea in terms of structural stability!

However... 4.2 cubic metres is like 2.5ish sq. metres I believe, so you don't need that much in terms of volume fallen. Your other issue is that if it's 900sq ft you're getting a lot of volume even with little depth, so (does math) if you go 8 cm deep, which is around 3", you've got 24 cubic metres of snow, which, see above, should be plenty?
 

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Reading over Cornflake's post, the snow in my anecdote was very fluffy, so it got squashed down a lot in the making of the snowman. (Lol, we didn't care. Beggers can't be choosers, and we got enough snow to make proper snowmen maybe three times through my childhood).

ETA - Also, it was melting as they worked, so they had a diminishing supply (it was gone by nightfall, leaving only the snowmen as proof it ever existed).
 
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Richard White

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Remember, piling all that snow up in one spot may or may not compromise the building structure.

Case in point. We had a large snowstorm a few years ago. The local parking garage was using snow plows to push the snow into a huge pile on the top floor so a crane could scoop/remove it. However, really wet snow weighs a lot more than you think. About 30 minutes before I arrived to work that morning, the snow pile got so large that it crashed through the reinforced concrete of the parking garage roof, taking out the 2nd floor with snow and falling concrete and piling up on the 1st floor - in the spot I normally parked. If I'd gotten to work earlier that day, that could have been my car under all that concrete and snow.

I still have a picture of the garage with the huge hole in it somewhere at home. It's funny now, not so funny at the time.
 

Chris P

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DougR: I'm getting different numbers. 900 square feet to a depth of five inches (0.4167 feet) gives me 375 cubic feet, which is about 10.6 cubic meters. Have I missed something?

Orianna: That's still enough for the snowman you want to build. I'm not trying to make your head spin with the numbers, but here's the logic if you need your character to explain himself. By plugging 375 cubic feet into the equation for the volume of a sphere, you could build one snowball 4.5 feet in radius, or 9 feet tall and 9 feet side to side.

In reality, because the widest part of the snowball would be 4.5 feet off the ground, you would need to make a cylinder. This is harder to calculate because you have two variables: radius and height. You can make a thinner cylinder taller, or a shorter one wider. To make your snowman 4.5 m (about 15 feet) tall, you will need to make him about 5.6 feet wide (2.8 feet radius).

It gets more complicated if you were going to make a 3-tier snowman. You would need to decide if you want the volume of each tier to step down in a 3:2:1 ratio (which because it's volume is going to look weirder than stepping down in radius), you would need to first decide if you were going do spheres or cylinders, then set up an equation where 375 cubic feet = (3 times the volume lower tier) + (2 times the volume middle tier) + (1 times the volume of top tier). If you are stepping down on radius, that's doable but will make your calculations more complicated. If you absolutely want numbers I can give them a whirl, but I'm kind of hoping you don't want numbers! :)
 

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DougR: I'm getting different numbers. 900 square feet to a depth of five inches (0.4167 feet) gives me 375 cubic feet, which is about 10.6 cubic meters. Have I missed something?

No, I screwed up the roof calculation.

My snowman calculations started with a base of 2m diameter, a middle 1.5m in diameter, and a top with 1.125m in diameter. That's 4.6 meters tall. The 2m base and 4.5m height were her specified parameters. With a total volume of 6.7 cubic meters of snow, her characters would clear off 63% of the roof.

The entire snowman would weigh 3549 pounds, and each ball would weigh 2218, 937, and 396 pounds respectively. They wouldn't be able to roll these balls, much less lift them.

Sorry for my bad maths. Earlier post deleted.
 
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blacbird

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Some real math: A cubic meter of water weighs right around a ton (2000 libs, 900 kg). A snowman base 2 meters in diameter would result in a sphere of packed snow amounting to ~3 cubic meters, or roughly 3 tons. Which doesn't count the weight of the smaller spheres on top of it. Packed snow would weigh somewhat less than water, but not a lot.

That would be a hell of a big and heavy snowman for somebody's roof.

caw
 

Chris P

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No, I screwed up the roof calculation.

My snowman calculations started with a base of 2m diameter, a middle 1.5m in diameter, and a top with 1.125m in diameter. That's 4.6 meters tall. The 2m base and 4.5m height were her specified parameters. With a total volume of 6.7 cubic meters of snow, her characters would clear off 63% of the roof.

The entire snowman would weigh 3549 pounds, and each ball would weigh 2218, 937, and 396 pounds respectively. They wouldn't be able to roll these balls, much less lift them.

Sorry for my bad maths. Earlier post deleted.

No worries about the math! I checked mine several times since I quite often forget to carry a one or square when I should cube.

The weight considerations everyone points out could enter into the OP's character's discussion if the character is as i-dotty as she says. However, if the roof could hold all that snow spread out, why couldn't it hold it if all in one snowman? The answer is a big duh, but I know people who would try to make that argument.

About compaction. If, just as a rough guess, my footprint in 5 inches of snow compacts it to about 1 inch, and if making a snowman compacts it to about the same density as a footprint, then we go from 375 cubic feet of snow down to one-fifth that: 75 cubic feet (while the weight stays the same!). Using Doug's dimensions to get to 4.6 m tall, that's 4.2 cubic meters for the base, 1.76 cubic meters for the middle, and 0.75 cubic meters for the top, that's 6.71 cubic meters total, which (since Google loves us and would never lie) comes to 237 cubic feet. Not nearly enough if I've estimated the compaction correctly.
 
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Al X.

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The bigger problem is that roofs in snow country are designed to withstand normal snow loads. They are not designed to withstand abnormal snow loads. Your epic snowman on the roof will, in reality, cause an epic structural failure of at least the roof.
 

Al X.

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Some real math: A cubic meter of water weighs right around a ton (2000 libs, 900 kg). A snowman base 2 meters in diameter would result in a sphere of packed snow amounting to ~3 cubic meters, or roughly 3 tons. Which doesn't count the weight of the smaller spheres on top of it. Packed snow would weigh somewhat less than water, but not a lot.

That would be a hell of a big and heavy snowman for somebody's roof.

caw

Point well taken but snow is -usually- in a non compacted form a fraction of the weight per volume of liquid water. I do agree with your bottom line conclusion.
 

jclarkdawe

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Roofs consist of rafters with a covering such as plywood over them. The rafters carry the load, with the plywood spreading that load onto the rafters. Failure can be either local (the plywood in between two rafters fail) and general (one or more rafters gives out). Failure can be either partial creating a leak, or total (a big freaking hole).

Normal snow load is relatively evenly distributed over the roof's area. A snowman, on the other hand, will take all of that weight and put it into a small area. Most likely you'd get a local failure, but with that sort of weight, you're probably going to get a failure. Snowmen are a lot heavier then snow that has not be compacted. Think about how a small starts with a larger mass then the packed snowball that you throw.

Without knowing the exact details of the construction of a roof, it's hard to know what it will support.

Jim Clark-Dawe
 

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Without knowing the exact details of the construction of a roof, it's hard to know what it will support.

Indeed.

Note also that this story takes place in London, a city I lived in for most of my life.

Snow very rarely falls in London. Not even in the outer reaches of London's suburbia. In all my suburban childhood we only had two snowfalls that remained for more than a few hours. When I worked in the centre of the West End I can only remember one day when it was snowing, and nothing settled.

Then you have to consider London's architecture: it's mostly very old buildings, all of which have pitched roofs, so building a snowman on top of one of those would be a proper challenge. Most of the apartment buildings are conversions of older buildings. The relatively few purpose-built ones tend to be a lot higher than just four floors; they still tend not to have flat roofs; and they don't usually allow residents access to those roofs.

We do tend to use metric measurements, but to be honest you're likely to hear both. However, housing isn't sold on the basis of square footage. It's usually advertised as having two bedrooms, three bedrooms, and so on, and then property details give the sizes of those bedrooms. We tend to use the word flats rather than apartments, but apartments is becoming more popular.

You'd be wise to work out where in London this block of flats is likely to be, as that will have a huge affect on what the building is like, what the area is like, and so on. Or to decide how you want the area to look, how affluent (or not!) you want it to be, then find a place that suits.

I love London. It's the most magical, wonderful city I've ever been to. I am so lucky to have lived there.
 

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Besides looking at the volume of raw material, remember that the fallen snow is fluffy, and you compress it to make a snowball.

Google shows me that the density of fresh snow can be anywhere from 20 to 257 kg/m³ — over an order of magnitude difference.

A hand-packed snowball is around 75% the density of water, which would be 750 kg/m³.

So you are looking at least at a 3:1 reduction in volume of the raw material, as high as 33:1, depending on the snowfall. My ancient recollection is about 5:1, which is in this range.
 

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I would second that for the sake of realism (assuming thats what you are going for, i dont know?? it could be elves and fairies building a snowman and then the snowfall statistics wont matter), 5" of snow in London would be absolutely miraculous and would not get slushy in the street as the city would grind to an absolute gridlocked halt. Nothing would move in terms of transport and most people wouldn't be out walking about either.

I would also guess that the roof would be far bigger than 900 sq feet which is only 30 foot per side, as in most "flats" there are more than 1 per floor. taking into account additional space for stairwell and lift etc it would be more likely to be a footprint of 2000 sq/feet.

Generally measurements in the UK are imperial, certainly in anyone older than 35 years old feet (in conversation a metric measure is often questioned and translated into imperial terms as it just sounds wierd to us), inches, pounds (weight), miles etc in common conversation and are still common enough in measurements of homes etc.
 

Orianna2000

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Wow, I wasn't expecting this many replies! Thanks, guys!

Wait, it doesn't snow in London? Are you serious? It's like, freezing there all the time! How can you not have snow?? I could've sworn I've seen it snow on Doctor Who.

More detail. The flat is narrow, but three stories tall, with one bedroom, a living area, and a bathroom on each level. So I think my 900 square feet is about right? I saw the building when I was last in London, but I didn't pay that much attention to the roof. One of the characters owns the building, so they will be allowed up there. I'm pretty sure the roof was flat, but I couldn't swear on it.

Also, I may have forgotten to mention that they aren't actually building the giant snowman. It's just that one of the characters is very anal about things and so he calculates the biggest snowman they could, in theory, make, given the amount of snow on the roof. The one they end up building is much smaller. Hope that clears things up!
 

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You might get a dusting of snow, which will have the same effect on the city as what you would imagine 3 foot of snow would.

maybe if the building is very narrow but from almost all flat blocks i know of, i have never seen one with 1 flat per story but i couldn't say for certain. your probably correct on flat roof as they are quite common. google earth might help with measurements f you can remember the building.

I would assume your character that owns the building to be ludicrously wealthy also.

I like this character already, says a lot about a person how "engineerily logical" people get about fun things.
 

lonestarlibrarian

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I don't know about London real estate, but in the US, my sfh's usually run about 700-900 sf for a 2/1. My 1/1's are usually around 500-600 sf. Once you hit 1000-1300 sf, you're usually in 3/1 and 3/2 territory.

With an apartment building, you'll need to set aside space for at least one stairwell (and possibly an elevator), and you'd probably compress things a bit for an urban mfh environment, unless it's an old building from an era when space was much more generous.

Looking at some London real estate sites (*looks at prices* *dies*), it doesn't seem like they describe the square footage much. Descriptions tend to be very vague-- a "nice-size bedroom, a fully fitted kitchen, and a fully fitted and tiled bathroom". Occasionally, if you find a site that has measured floorplans, you can do the math yourself. Oh-- here's a helpful one-- a kinda/sorta 2/1 is 742 sf; a 2/1 is 671 sf; a studio is 207 sf.
 

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Wow, I wasn't expecting this many replies! Thanks, guys!

Wait, it doesn't snow in London? Are you serious? It's like, freezing there all the time! How can you not have snow?? I could've sworn I've seen it snow on Doctor Who.

It very rarely snows in London. Not even in the suburbs. London is so huge and busy that it has its own microclimate, and it's far too warm for snow to fall or settle most of the time. As I said, I grew up in the London suburbs (in Hayes, if you're interested: it's near Heathrow Airport, so is nowhere near the centre of the city) and can only remember two times that the snow settled and lasted more than a few hours.

Yes, it does snow in London in Dr Who, but that's because the Doctor attracts anomalies. What happens around him can't be relied upon to happen when he's not there. And he's not often there, sadly.

More detail. The flat is narrow, but three stories tall, with one bedroom, a living area, and a bathroom on each level. So I think my 900 square feet is about right? I saw the building when I was last in London, but I didn't pay that much attention to the roof. One of the characters owns the building, so they will be allowed up there. I'm pretty sure the roof was flat, but I couldn't swear on it.

I don't suppose you know which street it was on, or have an address or anything? You could look at Google Maps. However, if it has a flat roof then that helps because it will almost certainly be a modern building, built since WWII--which means it will be smaller than the older buildings which surround it, as developers tend to have squeezed in as many buildings as they can as various sites have been redeveloped. So this would fit with your description of the building as narrow.

Also, I may have forgotten to mention that they aren't actually building the giant snowman. It's just that one of the characters is very anal about things and so he calculates the biggest snowman they could, in theory, make, given the amount of snow on the roof. The one they end up building is much smaller. Hope that clears things up!

Remember, too, that most roofs aren't made for people to walk around on. If you want them to go out onto the roof it's going to be uncomfortable for them: there won't be surfaces you can walk on, usually, and if there are they'll be relatively narrow walkways. Even "flat" roofs usually have a small slope on them to aid drainage, and the construction isn't usually good for walking about on--think of roofing felt rather than concrete, for example. Our buildings are a lot different to yours.
 

Orianna2000

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You might get a dusting of snow, which will have the same effect on the city as what you would imagine 3 foot of snow would.
When I was a kid, we lived in central California. It snowed one time, effectively shutting down the entire region. Schools closed, sending kids home halfway through the day. The only highway between our town and the closest city was shut down, so my mom was trapped there overnight. Everything came to a standstill. You'd think it was a blizzard, but years later, we looked at photos of the great snowfall, and it was no more than an inch and a half!

I would assume your character that owns the building to be ludicrously wealthy also.
Indeed they are.

I like this character already, says a lot about a person how "engineerily logical" people get about fun things.
Yes, I was quite tickled when I got the idea for this scene. I hope I can make it work, despite the apparent lack of snow in London. . . . It's also a pivotal moment, when one character realizes he's in love with the other.

With an apartment building, you'll need to set aside space for at least one stairwell (and possibly an elevator), and you'd probably compress things a bit for an urban mfh environment, unless it's an old building from an era when space was much more generous.
Have you seen the Victorian-era townhouses in London? I can't speak for all of them, of course, but the one we toured was quite cramped. The rooms were tiny, the bathroom was actually in the attic, there was no shower or bathtub, just a toilet and a basin/jug. I don't know how they managed during the era of hoopskirts and bustles!

It very rarely snows in London. Not even in the suburbs. London is so huge and busy that it has its own microclimate, and it's far too warm for snow to fall or settle most of the time. As I said, I grew up in the London suburbs (in Hayes, if you're interested: it's near Heathrow Airport, so is nowhere near the centre of the city) and can only remember two times that the snow settled and lasted more than a few hours.
We stayed near there the first time we visited London! This was over five years ago, yet my hubby actually remembers which tube station was closest to where we stayed . . . I envy his brain, sometimes.

Yes, it does snow in London in Dr Who, but that's because the Doctor attracts anomalies. What happens around him can't be relied upon to happen when he's not there. And he's not often there, sadly.
True. Sad, but true. I didn't see any Doctors at all while we were in London. Although, I did run into Billie Piper at the V&A gift shop.

I don't suppose you know which street it was on, or have an address or anything? You could look at Google Maps. However, if it has a flat roof then that helps because it will almost certainly be a modern building, built since WWII--which means it will be smaller than the older buildings which surround it, as developers tend to have squeezed in as many buildings as they can as various sites have been redeveloped. So this would fit with your description of the building as narrow.
I do have the address, but I'm almost afraid to look at Google Maps, in case I remembered wrong and my idea is completely impossible. It was an entire row of similar flats, identical, all up and down the street. I've no idea how old they might be, but I might be able to get my husband to look it up for me. (He's good at finding out real estate facts.)

Remember, too, that most roofs aren't made for people to walk around on. If you want them to go out onto the roof it's going to be uncomfortable for them: there won't be surfaces you can walk on, usually, and if there are they'll be relatively narrow walkways. Even "flat" roofs usually have a small slope on them to aid drainage, and the construction isn't usually good for walking about on--think of roofing felt rather than concrete, for example. Our buildings are a lot different to yours.
Another delusion shattered! Well, this place is going to have to be special, or else I'm going to have to make major modifications to the story, because I have another pivotal scene that takes place on the roof. One of the characters turns the roof into a garden, with boxes of dirt to grow plants in. And she's working up there when the other character comes home after a 14 month absence. I have the reunion scene all planned out, and I've no idea where else I could set it.

Don't you hate it when reality butts its ugly head into things that are none of its business?? :Headbang:
 

cornflake

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I'm interested that the roofs there on townhouses aren't generally flat or for walking, as the roofs here on brownstones are. Many of the brownstones are from the mid-late 1800s (so perhaps it's a later thing if the London ones are earlier), and have flat roofs people walk on, have gardens, patio furniture, etc. on. A friend of mine occupied the top couple floors of a brownstone a while back and we could not only go hang out on the roof, but walk across the roofs moving down nearly the entire block (there are little half-walls like knee high separating the buildings).

The buildings I'm talking about look like this.

Here's an aerial view of some.

This is a fancied-up roof (they put down planking)

And this is a kind of perfect view of a few and how they connect -- the orange-topped little walls are the dividers between those three buildings. The people on the left didn't put down anything, that's what's on most of the roofs, it's vaguely spongey. People in the center and to the right covered it.

All that said, don't a lot of London townhouses have back gardens? They do here, but I think they do there as well?
 

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Yes, I was quite tickled when I got the idea for this scene. I hope I can make it work, despite the apparent lack of snow in London. . . . It's also a pivotal moment, when one character realizes he's in love with the other.

If your setting is on the outer reaches of Greater London, rather than in the centre, it's going to be more believable. But otherwise I'm afraid you're going to struggle. This would be a major thing.

Have you seen the Victorian-era townhouses in London? I can't speak for all of them, of course, but the one we toured was quite cramped. The rooms were tiny, the bathroom was actually in the attic, there was no shower or bathtub, just a toilet and a basin/jug. I don't know how they managed during the era of hoopskirts and bustles!

By "townhouse" do you mean the terraced houses that are all over London? I've lived in several. Some are tiny and cramped but others are huge and wonderful.

I do have the address, but I'm almost afraid to look at Google Maps, in case I remembered wrong and my idea is completely impossible. It was an entire row of similar flats, identical, all up and down the street. I've no idea how old they might be, but I might be able to get my husband to look it up for me. (He's good at finding out real estate facts.)

Send me the address! I'd love to know where it is. Nosy, me.

Another delusion shattered! Well, this place is going to have to be special, or else I'm going to have to make major modifications to the story, because I have another pivotal scene that takes place on the roof. One of the characters turns the roof into a garden, with boxes of dirt to grow plants in. And she's working up there when the other character comes home after a 14 month absence. I have the reunion scene all planned out, and I've no idea where else I could set it.

Don't you hate it when reality butts its ugly head into things that are none of its business?? :Headbang:

If you're going to put plants on the roof like that you need to ensure the structure will cope with the added weight, and you'll usually need to add something like decking to make it work. If that's been done then you're good to go.

I'm interested that the roofs there on townhouses aren't generally flat or for walking, as the roofs here on brownstones are. Many of the brownstones are from the mid-late 1800s (so perhaps it's a later thing if the London ones are earlier), and have flat roofs people walk on, have gardens, patio furniture, etc. on. A friend of mine occupied the top couple floors of a brownstone a while back and we could not only go hang out on the roof, but walk across the roofs moving down nearly the entire block (there are little half-walls like knee high separating the buildings).

The buildings I'm talking about look like this.

Here's an aerial view of some.

This is a fancied-up roof (they put down planking)

And this is a kind of perfect view of a few and how they connect -- the orange-topped little walls are the dividers between those three buildings. The people on the left didn't put down anything, that's what's on most of the roofs, it's vaguely spongey. People in the center and to the right covered it.

All that said, don't a lot of London townhouses have back gardens? They do here, but I think they do there as well?

WANT.

Those houses are gorgeous.

Yes, most houses have gardens here, but blocks of flats tend to only have smallish communal spaces.
 

cornflake

practical experience, FTW
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Yeah, they're lovely -- most are divided up into separate apartments, but some are maintained as single-unit homes.

This is a renovated one; I don't like when people go that far. I think they're sort of destroying what's so great about them.

This is a restored one.

You can see the size of the back garden areas most have, and the roof in one.