teaching machines

Bathroom Angles

January 3, 2020 by Chris Johnson. Filed under public, twoville.

Our bathroom gave me a math problem, and I failed. Herein I document my failure and visualize the answer I should have provided.

When we bought our home, the bathroom had carpet but no baseboard. This combination makes painting the walls difficult. As we are preparing to paint, I decided to install some baseboard. Measuring, cutting, and hammering… These seemed like things I could do.

The only gotcha is that one of the corners between walls isn’t a 90-degree angle. It’s a 45-degree chamfer. I knew that I wouldn’t be able to just put two baseboards with perpendicular ends together at this corner. I’d end up with a join like this:

  

    with viewport
  corner = [-2.3, -3]
  size = [5.6, 3.3]

to board(angle, color, bend, cut)
  with path()
    color = color
    opacity = 0.7
    closed = true
    with turtle()
      position = :zero2
      heading = angle
    move().distance = 3
    turn().degrees = bend
    move().distance = 1
    turn().degrees = bend
    move().distance = 3 - 1 / tan(90 - cut)
    turn().degrees = bend
  
board(0, :red, -90, 0)
board(-135, :magenta, 90, 0)
    
    
    
  
  
    

    with viewport
  corner = [-2.3, -3]
  size = [5.6, 3.3]

to board(angle, color, bend, cut)
  with path()
    color = color
    opacity = 0.7
    closed = true
    with turtle()
      position = :zero2
      heading = angle
    move().distance = 3
    turn().degrees = bend
    move().distance = 1
    turn().degrees = bend
    move().distance = 3 - 1 / tan(90 - cut)
    turn().degrees = bend
  
board(0, :red, -90, 0)
board(-135, :magenta, 90, 0)
  
  expand

The red board and magenta boards overlap, which is a physical impossibility.

I could have them meet at the front instead, like this:

  

    with viewport
  corner = [-3, -2.4]
  size = [6.5, 3.8]

to board(angle, color, bend, cut)
  with path()
    color = color
    opacity = 0.7
    closed = true
    with turtle()
      position = :zero2
      heading = angle
    move().distance = 3
    turn().degrees = bend
    move().distance = 1
    turn().degrees = bend
    move().distance = 3 - 1 / tan(90 - cut)
    turn().degrees = bend
  
board(0, :red, 90, 0)
board(-135, :magenta, -90, 0)

    
    
    
  
  
    

    with viewport
  corner = [-3, -2.4]
  size = [6.5, 3.8]

to board(angle, color, bend, cut)
  with path()
    color = color
    opacity = 0.7
    closed = true
    with turtle()
      position = :zero2
      heading = angle
    move().distance = 3
    turn().degrees = bend
    move().distance = 1
    turn().degrees = bend
    move().distance = 3 - 1 / tan(90 - cut)
    turn().degrees = bend
  
board(0, :red, 90, 0)
board(-135, :magenta, -90, 0)

  
  expand

But this leaves an unsightly gap, which is an aesthetic impossibility.

My intuition kicked in and told me I needed a 45-degree cut on the board that was up against the chamfer. I obeyed and unwittingly birthed a monster. The boards aligned at the ends, but the fronts didn’t match up at all.

  

    with viewport
  corner = [-2.3, -3]
  size = [5.6, 3.3]

to board(angle, color, bend, cut)
  with path()
    color = color
    opacity = 0.7
    closed = true
    with turtle()
      position = :zero2
      heading = angle
    move().distance = 3
    turn().degrees = bend
    move().distance = 1
    turn().degrees = bend
    move().distance = 3 - 1 / tan(90 - cut)
    turn().degrees = bend
  
board(0, :red, -90, 0)
board(-135, :magenta, 90, 45)

    
    
    
  
  
    

    with viewport
  corner = [-2.3, -3]
  size = [5.6, 3.3]

to board(angle, color, bend, cut)
  with path()
    color = color
    opacity = 0.7
    closed = true
    with turtle()
      position = :zero2
      heading = angle
    move().distance = 3
    turn().degrees = bend
    move().distance = 1
    turn().degrees = bend
    move().distance = 3 - 1 / tan(90 - cut)
    turn().degrees = bend
  
board(0, :red, -90, 0)
board(-135, :magenta, 90, 45)

  
  expand

After some reflection, I realized that I was lining up the side of a 45-degree right triangle with the hypotenuse of a 45-degree right triangle, and they would necessarily not be of the same length.

To ensure that both ends had the same length, I would need to cut each at the same angle. In this case, I needed to split those 45 degrees between the two boards, cutting each with a 22.5-degree bevel.

  

    with viewport
  corner = [-2.3, -3]
  size = [5.6, 3.3]

to board(angle, color, bend, cut)
  with path()
    color = color
    opacity = 0.7
    closed = true
    with turtle()
      position = :zero2
      heading = angle
    move().distance = 3
    turn().degrees = bend
    move().distance = 1
    turn().degrees = bend
    move().distance = 3 - 1 / tan(90 - cut)
    turn().degrees = bend
  
board(0, :red, -90, 22.5)
board(-135, :magenta, 90, 22.5)

    
    
    
  
  
    

    with viewport
  corner = [-2.3, -3]
  size = [5.6, 3.3]

to board(angle, color, bend, cut)
  with path()
    color = color
    opacity = 0.7
    closed = true
    with turtle()
      position = :zero2
      heading = angle
    move().distance = 3
    turn().degrees = bend
    move().distance = 1
    turn().degrees = bend
    move().distance = 3 - 1 / tan(90 - cut)
    turn().degrees = bend
  
board(0, :red, -90, 22.5)
board(-135, :magenta, 90, 22.5)

  
  expand

This looks so much better.

One of the reasons I got into computers is so that I could play around with ideas without wasting materials on my mistakes. If only I’d done so here. This is what miter been.