This is an empirical method - plotting actual points on the wall over a minimum of 6 months. There exist mathematical tools to do this for plane surfaces, but any errors in orientation, surface finish and measurements will result in errors in reading time. Actually plotting the points washes out all those variations and gives a remarkably accurate sundial.
Pick a site with good solar exposure - a true East-West wall is ideal, but small variations will still work. You just lose more morning or afternoon readings as the wall varies more from the E-W line. Also, pick an area with few shading problems - walls, fences, trees and such. The Gnomon will need to be mounted near the top of the wall, in a place that won’t get bumped or damaged.
In the end though, pick a place you can see and want a dial, and work around the issues. There are a lot of beautiful dials tucked away on the outsides of attics or in corners - just think creatively.
This should be a rigid triangle that matches your latitude when installed. A right triangle is easiest to work with, but for artistic effect any shape could be used. The only rule is that the top edge needs to point towards your celestial pole. That way it is parallel to the Earth's axis and casts the best shadow for a dial.
The exposed height of your wall will establish the size limit for the Gnomon. Use the table below for a first estimate based on your latitude. The table is scaled to a wall 1 unit high. It will work for feet, meters, inches, centimeters or whichever units are used. Multiply the length factor by your available height and that will give a maximum length the Gnomon can project from the wall and not cast a shadow beyond the wall and onto the ground.
Pull data for the Equation of Time (EoT) for the months you will be recording points.
A good site is: http://www.ppowers.com/EoT.htm.
This is the correction for each day between clock time and Solar time. For example, December 21 of 2018 has a correction of 2:21. The positive sign means that the Sun is AHEAD of clock time. Throughout that day you will record the marks 2 minutes and 21 seconds BEFORE the clock time. The 10 o’clock mark will be at 9:57:39, the 11 o’clock mark at 10:57:39, and so on.
Clear your calendar for the 21st of each month - you really can't fake the data. Verify the dates for solstices & equinoxes - due to the leap year variations they might actually be on the 20th or 22nd for the year you are working. No sense making a dial if it isn't as accurate as possible ...
Actually, a simplified dial could be made with just four days data - the two solstices and the two equinoxes. The data collection process is the same for each day you are able to spend working on it.
Whatever your plan, get a good source for accurate time, and a fresh Sharpie. Cellphone time is generally very good, but you should verify yours before starting out. Use this link from the US government:
The points will be marked at the time corrected for our elliptical orbit by the equation of time, NOT just clock time. For example, on June 21st, 2018 the correction is -1 minutes 48 seconds. The negative sign means the dial is SLOW relative to clock time, so you mark that much AFTER clock time.The 12:00 mark will be made 12:01:48. Each mark that day should be made with the same offset. At a minimum, each hour your dial is illuminated should be marked. I chose to mark every 15 minutes on the key dates, so really had to stay close by.
If you marked at clock time each day, the points would plot out a figure-8 shaped Analemma through the course of the year. The curves would be centered on the average hour lines this method produces. This dial would actually be directly readable without a conversion but would take significantly more time & effort to plot. I do recommend plotting clock time for the central hour line to give a visual correction factor. Marking about once per week gives enough resolution to connect the points. See the black Analemma on the photos of the full dial.
Each month the routine is similar - set alarms a couple minutes ahead of the corrected times and work on other projects nearby. When the alarm rings, go out and mark the point at just the right second. On the equinoxes (September 22 and March 20 usually) the shadow should track a dead straight line and prove that the Sun is centered on the plane of the ecliptic. Once the data is marked from one Solstice to the next you will have the full boundaries of the dial identified.
Once your data is complete, the artistic part of the project is ready to go. All I can say here is to look at lots of other dials, but don't copy any of them. You should borrow good ideas but make it your own sundial. There are centuries worth of great examples out there to work from.
Be sure to share your work - I'd love to see what you come up with!