You can find the Affinity Photo equations filter in the filters menu and distort.

Access the settings and extend modes etc for the filter) and set x = x*5 and set y=y*5 and set extend mode to wrap and click apply.

You can currently use Cartesian (x,y) and polar (r,t) If you want to create radial design then the polar is the best option, but I generally stick with the Cartesian and use x and y. You cannot mix the r and t with the x and y (sadly).

Extend modes of zero, full, repeat, wrap, mirror. Wrap and mirror are excellent for tiling (mirror is the best) but zero is for showing a transparent background, set it to full for a white background and set it to repeat if you want to stretch the edge pixels (really don't like that effect). You can also use rx and ry in the entries instead of x and y but beware that the filter flips the vertical for some reason when it uses that, not sure why.

You only have three parameters which have a range of 0, 1 and they are a, b, c. Use those in the edit fields for x and y, so you can have x * a and x * a * 10 etc. Sadly, there is no edit field for fine tuning the value but if you want to do that, use decimals such as 0.5 * a or 0.05 * a or 100 * a etc

OK, a quick one to start with the editor. Perhaps you want your image to fill the screen 16 times in a grid 4 x 4 then you can do this by using the Cartesian and x*4 and y*4 for the x= and y= values, so if you have a document sized 1000 x 1000 then the image is reduced into a size 250 x 250 and then this is placed into the 16 sections of the original image. To get the images all up the right way then use the wrap extended mode. If you use the mirror then you will see the images mirrored. Repeat just places it in the corner. Full reduces it to the corner but with the background color and the zero the image in the corner with transparent background. It can be applied again and can also be used with layers to create interesting combinations of wrap and mirror etc Say you want to do the same to create 144 copies of the image then use 12 * x and 12 * y. Of course, you don't have to use the same for both of the x and y, you can use 12 * x and 6 * y etc

Will add to this list as I find out more and I am using p1, p2 etc for the parameter 1 etc but please note, p1 cannot be actually be used. In some cases, the value is definitely for x and y or a etc

noise (p1)

abs (p1)

sin (p1)

max (p1,p2)

lerp (x,y,a)

sqrt (p1)

oscsin (p1, p2)

etc

The examples generate a variety of different visual effects but the results will depend on the image and the layer. All the ones below are for a full image and no selection etc. You can also replace the values such as 5000 with a or b etc but you should also remember to apply a suitable factor to the a and b to allow it to actually reach 5000. Please feel free to use any of the below in your projects. All are for the wrap extend mode. First column, Polar or Cartesian. Second column is for x and third column is for y.

Polar | r | t*12 |

Cart | x*sin(x) | y*cos(y) |

Cart | x+150 | y+150 |

Cart | x+5000/x | y+5000/y |

Polar | r/sin(r/2) | t |

Polar | 4*r/t | t |

Polar | r*average(r,t)/20 | t |

Polar | r*sin(r/4) | t |

Polar | 500*noise3(r/30,5*t) | t |

Cart | x-30*sin(200*noise3(x/10,y/10) | y |

Cart | x+y | y |

Cart | x+y | y-x |

Cart | y-x | y-x*2 |

Cart | 3*x + 10*min(x,y/2) | 3*y + 10*min(x,y/2) |

Cart | x*sin(y)*cos(x)-x | y |

Cart | x*sin(y)*cos(x)-x | y*sin(x)*cos(y)-y |

Cart | 4*y-x*sin(y)*cos(x)-x | 4*x-y*sin(x)*cos(y)-y |

Polar | r*sin(t*r-t)*cos(5*r)-r | t*sin(t*r-r)*cos(t)-t |

Polar | r-300*Tan(r/3) | t+200*Tan(2*sin(t*2) |

Cart | x+Tan(x*2) | y+Tan(y*2) |

Cart | x+x*sin(y-x/2/2) | y |

Cart | x*sin(sqrt(x*x+y*y)) | y |

This uses one of the more obscure functions, oscsin. The oscsin is an oscillator and combines with a sine wave to create all kinds of amazing imagery. The input includes two parameters, x and y or in this case, a relative x and a relative y. I prefer to use the relative option as you can move the layer and re-position the image. It uses the underlying image. Create an image layer, fill it with color, noise, gradients etc and then apply the equations filter. The oscsin input can be varied, it does not need to be just rx or ry, you can divide the rx and ry by a constant as well as a custom input (a, b, or c). The oscsin can be used on its own but to create more interesting combinations, use the oscsin (factor) minus oscsin (another factor) or perhaps instead of minus, use multiply. To create the grid ripple effect, use

X = 1000*a*(oscsin(rx/(b*1000),ry/(b*1000))-oscsin(rx/(c*1000),ry/(c*1000)))

Y = 1000*a*(oscsin(rx/(b*1000),ry/(b*1000))-oscsin(rx/(c*1000),ry/(c*1000))) in the two fields. You will need to use the a, b, and c. Set extend mode to mirror.

You can use the filter as is. You can also fade the result. Go to the layer menu and fade command. The opacity as well as the blending mode for effect can be added. Use difference, darken, lighten etc and create even more amazing imagery.

Shapes can be distorted in infinite ways using the effect. Select any shape via the shape tools or use the pen tool to create your own vector layer. Go to the filter via the distort category. Copy the formula into the field for the x and y and set the extend method, mirror works well. If you have been using the filter with a normal layer, the a, b, and c settings may need a little tweaking and experimentation. You can also vary the constants in the formula so if 1000 is used, 2000 may be even better with the shape. Once you are happy with the design, press apply. The layer should have transparency and you can add layer effects to the design such as adding an outer shadow or bevel. Other filters can be used.

The effect and other steps can be recorded in a macro and saved to the library. You can also use the parameters feature in the macros to change the settings as you apply it in playback.

You can use a variation of oscsin(a*rx/100,c*50)*ry in your edit fields and create all kinds of unusual sin wave themed transformations of an image and especially helpful to use the extend mode to mirror. You cannot store the effect as a presets sadly but a workaround is the macro panel where the steps can be saved to a macro and then played back at any time. You can import and export macros so the files can be sent to others. You can also change the parameters when using macros, so you don't have to use the same a, b, etc when you re-apply it.

You can use the effect once or apply it multiple times such as repeat scaling etc to really zoom into an image or shrinking an image etc.