Tp create the Affinity Photo patchwork effect, go to the layer menu and new layer and then go to the gradient tool and add a gradient (you can tweak it to add more stops etc to create a super colorful gradient.
With your basic gradient, you can now go to the filters menu and distort and equations. The panel has a number of options such as coordinate system and angular (I generally don't change those so Cartesian and radians) and the x and y fields and that is it. You can change the parameters and they relate to the fields marked as a and b and c. Sadly there is no presets feature and only three parameters are available. The x and y need to be entered manually
For x you can enter (w * a * sin(b * rx/10)) + (w * a * sin(b * rx/20)) + (w * a * sin(b * rx/30)) and for y you can enter
(h * a * sin(b * ry/10)) + (h * a * sin(b * ry/20)) + (h * a * sin(b * ry/30))
The w is the width of the image, h is the height and the parameters are a and b and c and the rx and ry are relative positions of the pixels and because it is rx and ry you can move the panel etc, if it was just x and y then the image would be frozen unless you added in some shift feature. The extend mode to "Mirror" or "Wrap" (at the bottom of the panel)
You will notice that c is not used as a parameter but the patchwork can easily be modified by adding the c parameter into the x or y or both. Decreasing b zooms into the design. x can be turned into (w * a * sin(b * rx/10)) + (w * a * sin(b * (rx-ry)/c)) + (w * a * sin(b * rx/c)) and y can be turned into (h * a * sin(b * ry/10)) + (h * a * sin(b * (ry-rx)/c)) + (h * a * sin(b * ry/c)) so where we had rx and ry and 20 and 30 there is now a value of c as well as introducing a rx - ry and ry - rx into the equation to create a diagonal effect. A change of the parameters can radically change the end result
It can be applied to a layer but it can also be applied to multiple layers. It can also be applied to different channels such as the red channel etc via the channels panel found in the view and studio menu.
You can also use the result combined with other filter effects such as deform or mirror. As the equation filter does not include presets, a good way to work with the equation filter is to combine and remember effects via a macro and that can be done via the view menu and studio and macros. Deform etc can be applied multiple times to the same design. You can also combine it with the original gradient by using the layer menu and fade command and using different blending modes and / or opacity.
It can also be used with selections as well, either the standard rectangular marquee or freehand selection tool or the select menu tonal etc selections.
Sometimes the simple act of putting sqrt () around all the entire function can completely change the end result. Affinity Photo comes with a whole long list of functions that can be used with the tool. You can also use r x + ry and rx - ry etc to create diagonal designs and circular effects, you can also tweak other constants and add a c parameter to the equation. All kinds of different changes can be made such as just adding a rx / 3 so the rx diagonal is slightly different from the rx one.
The macro panel is super useful as you can simply record this action with the formula for an instant effect with any image (you can save the macro the library panel for future use), You can then apply it multiple times via a click of the play in the macro panel. Record the creation of the effect as well as the initial setting up of the parameters a b c and also the mirror / repeat etc extend mode. After you apply the effect, stop the recording and then go to the little gear on the entry and click that and you will see parameter a / b / c and extend mode. You need to click the little eyes on the left of each and give them a name etc. If they have the eye set to ON then when you playback the macro then a panel will pop up and you can then change the parameters. Sadly, you cannot at this point change the formula.
If you have set up the parameters, each and every time you run the effect / macro a panel will pop up and you will be able to change the value of the a and b etc and you will see a live update of the effect. You can also save this macro and parameters to the library by clicking the add to library button on the panel. You will then see a panel asking for a category and a name and clicking apply, you will see it in the library panel. When you now run it via the library, the panel will pop up for the parameters. Now this means you can run it with channels as well as layers and also use it with the repeat command as well as via the fade command found in the layer menu.
You can also apply the effect multiple times on different layers and combine them using blending modes via the layers panel such as darken or darker color etc to create some truly odd effects