Crochet Flamm’s Paraboloid

Yes, this is how I spend my free time.

Credit: Allen McC

Following on from my previous post about Schwarzschild Black Holes, I decided to crochet a little Flamm’s Paraboloid. This was not an easy task, as I had to fudge the numbers since crochet works in discrete stitches; you can’t have 3/7 of a stitch!

The general equation for Flamm’s paraboloid is as below, you can think of w and r as y and x on a graph:

And below is a graphical drawing of it, I’ve used r_s = 1, since 1 is nice and simple (you’ll notice how the curve also starts at r=1. Flamm’s paraboloid describes spacial curvature outside a black hole, so it would evidently start at the event horizon. Monkey brain didn’t realise that until later!)

Now this graph is 2D, but you can image if you rotated that line around the w axis it’d look like the colourful diagram at the top.

The drawing we’ve made has a variable gradient, which means we need to guesstimate and chop up the line in small segments. Crochet can’t account for infinitesimal changes in gradients, so we do the best we can!

Creating The Pattern

I found points with equally spaced apart y values (so y=0, 0.5, 1, 1.5 etc) and noted their corresponding x coordinates. As the radius r is linearly correlated with the circumference, we can just take the x coordinates as they are and use them as a scale factor as we make more rows. Basically, when the r coordinate increases, the circumference increases by the same amount, which is useful for crochet as we’re just making a lot of circumferences stacked on top of each other.

I will let 1 row = a y interval of 0.5. My r coordinates were as below:

  • 1
  • 1.063
  • 1.25
  • 1.563
  • 2
  • 2.563
  • 3.25
  • 4.062
  • 5

To make a 3D item, I won’t just make 1 column of stitches. I will make 10 and join them in the round. Multiplying the above x coordinates by 10 and rounding them to the nearest integer will give me the amount of stitches I need per row:

Row 9 complete!
  • Row1: 10 stitches
  • Row 2: 11
  • Row 3: 13
  • Row 4: 16
  • Row 5: 20
  • Row 6: 26
  • Row 7: 33
  • Row 8: 41
  • Row 9: 50
  • Row 10: 61

Interestingly enough, to get to row 2 I increase by 1, then to row 3 I increase by 2. Then 3, then 4, then… not 5? It seems that to make Flamm’s paraboloid I nee to increase my increases by 1 for each row, but I skip multiples of 5. For row 9 I increase by 9, but for row 10 I need to increase 11. Interesting!

The Pattern

I will use half double crochet (hdc), but this pattern is customisable! The standard abbreviations apply here.

Foundation Round: I chained 10 and made a slip stitch into the first chain to join in the round.

Round 1: Chain 2, make 1 hdc in each chain, slip into the first hdc. (10)

Round 2: Chain 2, make 1 hdc in each of next 4 chains, make 2 hdc in next stitch, hdc in each of the rest of the stitches, slip into the first hdc. (11)

Round 3: Chain 2, make 2 hdc in next stitch, hdc in each of next 5 stitches, make 2 hdc in next stitch, hdc in each of the rest of the stitches, slip into the first hdc. (13)

Round 4: Chain 2, * make 2hdc in next stitch, hdc in each of next 3 stitches, * repeat 2 more times, slip into the first hdc. (16)

From now on make increases evenly spaced throughout the round.

Round 5: Chain 2, hdc in each stitch while making a total of 4 increases, slip into the first hdc. (20)

Round 6: Chain 2, hdc in each stitch while making a total of 6 increases, slip into the first hdc. (26)

Round 7: Chain 2, hdc in each stitch while making a total of 7 increases, slip into the first hdc. (33)

Round 8: Chain 2, hdc in each stitch while making a total of 8 increases, slip into the first hdc. (41)

Round 9: Chain 2, hdc in each stitch while making a total of 9 increases, slip into the first hdc. (50)

Round 10: Chain 2, hdc in each stitch while making a total of 11 increases, slip into the first hdc. (61)

You can keep on going as long as you want, but I stopped here. Below is the finished piece:

That was an interesting thing to learn about. Let me know how you found it, and feel free to give suggestions on what I could crochet next!

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